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Full text of "Logic, in three books; Of thought, Of investigation, and Of knowledge;"

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LOTZE'S SYSTEM OF PHILOSOPHY 
PART I 

LOGIC 



Logic. Vol. II. 



a 



HENRY FROWDE, M.A. 

PUBLISHER TO THE UNIVERSITY OF OXFORD 




LONDON, EDINBURGH, AND NEW YORK 



'Eb.£ Clarcnboit press Series 

LOGIC 

IN THREE BOOKS 

OF THOUGHT, OF INVESTIGATION 
AND OF KNOWLEDGE 

BY 

HERMANN LOTZE 

ENGLISH TRANSLATION, EDITED BY 

BERNARD BOSANQUET, MA. 

FORMERLY FELLOW OF UNIVERSITY COLLEGE, OXFORD 

Second Edition, in Two Volumes 

Vol. II 



AT THE CLARENDON PRESS 
1888 

[ All rights reserved ] 



Digitized by the Internet Archive 
in 2014 



https://archive.org/details/logicinthreeboo02lotz 



CONTENTS OF VOLUME IL 



BOOK II. 
Applied Logic. 

CHAPTER VI. 

FALLACIES AND DILEMMAS. 

PAGE 



240. Premises must be true in order to prove a conclusion . i 

241. And must not covertly involve the conclusion ... 2 

242. Preposterous Reasoning confuses the principiatum as causa 

cognoscendi with the principium as causa essendi . 3 

243. Ambiguity of middle term mostly due to the confusion of a 

relative with an absolute truth ..... 4 

244. Illustration of the above from moral precepts, all of which 

have their exceptions ....... 5 

245. As have also mechanical formulae, which become unmeaning, 

when pushed to extremes 7 

246. Fallacies of too wide or too narrow definition . . 10 

247. Fallacy of incomplete explanation illustrated by the popular 

idea that lapse of time destroys motion . . . .10 

248. Incomplete disjunction the cause of much philosophical and 

other onesidedness 12 

249. The fallacy in Zeno's paradoxes about reality of motion . 14 

250. Examples of classical dilemmas stated and explained . . 17 

CHAPTER VII. 

UNIVERSAL PROPOSITIONS AS DERIVED FROM PERCEPTIONS. 

251. Inductive methods are based on results of deductive Logic . 22 

252. Connexions of elements revealed in sensible experience are 

mostly impure . . . . . . . 23 



vi TABLE OF CONTENTS. 

PAGE 



253. The universality of a pure connexion or its character as a, law 

of nature guaranteed by the law of Identity ... 24 

254. The raw matter of Inductions consists not of passive impres- 

sions but of perceptions already articulated by thought as 
subject and predicate and ranged under general concep- 
tions .......... 25 

255. They are so ranged by an incomplete analogy, based on a 

distinction of essential from non-essential remarks, which 
logical theory cannot assist . . . . . .28 

256. In reaching universal inductions we must argue ad subaltern - 

antem 31 

257. The truths of Geometry are universal because the diagram is 

used as a symbol only of our conception . . -33 

258. The highest inductions not categorical but hypothetical 

judgments 35 

259. Terms which are exclusively cause and effect of each other 

are related as ground and consequent . . . -37 

260. Experiment merely subsidiary to observation and has no 

peculiar virtue of its own 38 

261. Typical cases of the relation in which two phenomena Cand 

E may stand to one another 40 

(1) C and E Co-Exist always. 

(2) C and E frequently concur. 

(3) Absence of C not involving absence of E. Criticism 

of the canon 4 cessante causa cessat et effectus.' 

(4) Presence of C not involving presence of E. Differ- 



ence of relation of cause and effect and of ground 
and consequent. 

(5) Absence of C involving absence of E. 

(6) Presence of E involving presence of C. Criticism 

of Newton's canon ' effectuum naturalium ejusdem 
generis eaedem sunt causae.' 

(7) Absence of E involving absence of C. 

262. Whether the phenomenon C is or only contains the cause of 

E can only be decided by analysis of both into their 
elements and observation of which elements of the one 
involve which elements of the other. Typical examples 
of such analysis 52 

263. The exact nature of the causal nexus inferred from any of 

the above relations to exist between C and E can only be 
apprehended by observation of the quantitative changes 
they cause in one another. Examples of such quantitative 
correspondences 60 



TABLE OF CONTENTS. 



vii 



CHAPTER VIII. 

THE DISCOVERY OF LAWS. 

PAGE 



264. Science not content with discovering a mere connexion be- 

tween two phenomena seeks to know the law of this 
connexion ......... 67 

265. Laws of nature are universal hypothetical judgments and not 

assertions of universal matters-of-fact .... 68 

266. A law expresses an objective and intelligible connexion of 

phenomena, a rule is a mere subjective method of thought 71 

267. The ultimate criterion of sense-perception to be found in 

sense itself ......... 73 

268. Facts as they appear are not only relative to one another but 

to the standpoint of the observer, and must therefore be 
grasped as projections of ulterior and truer facts . . *j6 

269. A law always transcends the given, being an extension to 

cases not given of what holds good within the given. A 
truly universal law is not a demonstrable truth . . 79 

270. Laws based on statistics are mostly partial truths ... 84 

271. The law which prima facie best fits in with observed facts 

need not therefore be the truest expression of their inter- 
connexion ......... 85 

272. Simplicity no guarantee for the truth of a law. The simplest 

law only preferable where it is the sole conceivable one . 87 

273. A postulate lays down the conditions under which alone the 

given appearance is conceivable. A hypothesis is a sug- 
gestion of conceivable facts fulfilling the demands of the 
postulate and so explaining the appearance. A fiction 
views the given as an approximate realisation of a known 
law, in the absence of a known law to which it can be 
simply referred 90 

274. Rules for framing of hypotheses not to be laid down before- 

hand, but none to be rejected because beyond reach of 
refutation if false 94 

275. Hypotheses must satisfy their postulates and supply the 

conditions of the appearances to be explained 97 

276. An old hypothesis not to be hastily set aside but modified to 

suit the new and discrepant facts ..... 99 

277. A hypothesis must limit itself to asserting what is possible, 

i. e. what can be conceived or pictured as matter-of-fact . 101 



viii 



TABLE OF CONTENTS. 



CHAPTER IX. 

DETERMINATION OF INDIVIDUAL FACTS. 

PAGE 

278. In determining facts which transcend the immediate im- 

pression we must be guided by probability . . .104 

279. In view of the complexity of things a principle of explanation 

must not be too simple and abstract . . . .105 

280. And on the other hand it must involve as few presuppositions 

as possible. Positive evidence preferable to negative . 107 

281. The mathematical determination of chances assumes that 

they are all equally possible, but that one of them must 
occur 109 

282. (1) Mathematical chance no positive prediction of events. 

It measures our expectation of their occurrence . 113 

(2) theory of composite chances. 

(3) dependent chances, 

(4) probability of alternative causes. 

(5) probability of an event's recurrence. 

(6) mathematical expectation. 

(7) moral expectation. 

283. Calculus of chances not only presupposes the laws of all 

calculation such as law of Identity and doctrine of dis- 
junctive judgment, but also an ordered universe of 
interdependent events 127 

284. Mathematical chance is our subjective expectation of an 

event, and not a permanent property thereof. The result- 
ing chance improbable only as compared with the sum 
of its alternatives, not as compared with any one of them 130 

285. Success of attempts made to test by experiment the calculus 

of chances . . . . . . . . 133 

286. Such successful results not fraught with intelligible necessity, 

but the result of constant conditions operating among 
variable ones, which in the long run neutralise each other 135 

287. Use of the calculus in cases where constant and variable 

causes of an often repeated event are unknown. Nature 

of so-called statistical laws 139 

288. Use of the calculus in determining the probable accuracy of 

our observations of magnitude. The method of the least 
sauares . . . . . . . .. . • ] 4 2 



TABLE OF CONTENTS. 



ix 



CHAPTER X. 

OF ELECTIONS AND VOTING. 

PAGE 

289. Conditions presupposed by a logical treatment of the 

problem of expressing a collective will . . . .148 

290. Defects of absolute majority 149 

291. The weight of votes. A majority of majorities may be a 

minority of the whole constituency ..... 149 

292. Voting so as to express intensities of Volition . . .153 

293. Election by elimination 157 

294. When order of putting proposals to the vote is important . 159 

295. Rejection of innovations as such. 'Order of the day' . 161 

296. Amendments and substantive motion. Order of putting pro- 

posals to the vote . .162 

BOOK III. 

On Knowledge (Methodology). 

INTRODUCTION. 

297. Analytic and Synthetic methods practically inseparable . 166 

298. Correspond respectively to Investigation and Exposition; are 

more general than ' methods ' of applied Logic . .169 

299. But applied Logic, like common thought, rests on untested 

bases . . . . . . . . . .170 

300. And so does science as we have it 171 

301. Methodology however as treatment of Knowledge is enquiry 

into sources of certainty . . . . . . .173 

CHAPTER I. 

ON SCEPTICISM. 

302. Scepticism presupposes Truth and Knowledge . . .176 

303. But doubts whether our Knowledge is Truth. Descartes . 179 

304. This doubt invoves the assumption of a world of things which 

our thought should copy . . . . . .182 

305. But any decision postulates the competence of thought . 184 

306. Which can only be guided by conceptions in our minds . 185 

307. Our delusion could only be revealed by fresh knowledge . 187 

308. Which must be related to the old. Things are not knozvledge 

of things . . . . . . . . .189 



X 



TABLE OF CONTENTS, 



PAGE 



309. That Things may not be what they seem, as a mere general 

doubt, is self-contradictory . . . . . .192 

310. Sceptical arguments in Sextus Empiricus . . . .193 

311. They involve the above difficulties ..... 196 

312. Error in ' we only know phenomena ' 198 

CHAPTER II. 

THE WORLD OF IDEAS. 

313. Genesis of Plato's doctrine of ' Ideas ' ..... 200 

314. The Ideas as Universal conceptions 202 

315. Possible knowledge of Ideas apart from question of Things . 204 

316. Distinction between Existence, Occurrence, Validity . . 206 

317. Confusion of Existence and Validity in case of the Ideas . 210 

318. Ideas in what sense eternal, and independent of things . 211 

319. Aristotle on the Ideas. His universal too is ovcria . .214 

320. Modern counterparts of the Ideas. Validity a difficult notion 216 

321. ' Ideas impart no motion' criticised; importance of Judg- 

ments 218 



CHAPTER III. 

THE A PRIORI AND THE EMPIRICAL METHODS. 



322. Judging of knowledge by our notions of its origin an illusion 223 

323. Attempt to find a starting-point for knowledge. ' Cogito, 

ergo sum* . . . . . . ... .226 

324. Innate Ideas ; but are they true? 229 

325. Action of one thing on another implies Spontaneity in order 

to Receptivity . . . . . . . . 231 

326. Nature of mind is contributory in all elements of knowledge 232 

327. Both in simple Perception and in such ideas as that of causal 

connexion . . . . . . . . 234 

328. External reality must be criticised on ground of knowledge . 236 

329. Universality and Necessity as marks of a priori knowledge . 239 

330. Universal validity not derivable from repeated perceptions 

alone .......... 241 

331. There may be spurious self- evidence, which is tested by 

thinking the contradictory 243 

332. Use of psychological analysis in establishing first principles 246 

333. Even modern Psychology hardly helps Logic . . .248 



TABLE OF CONTENTS. 



CHAPTER IV. 

REAL AND FORMAL SIGNIFICANCE OF LOGICAL ACTS. 

PAGK 

334. Thought must have some Real significance . . . .252 

335. Comparison and distinction as acts resulting in Relations . 254 

336. Thought is symbolic and discursive ..... 256 

337. How can a relation of ideas be objective . . . -259 

338. Only as independent of individual mind. The case of Th ings 260 

339. A universal cannot be realised, but has objective validity . 264 

340. Nominalism and Realism confuse Existence and Validity 267 

341. The Reality of general notions is only validity . . . 268 

342. Conception not akin to object in structure, but in net result . 270 

343. Degrees of subjectivity in kinds of Judgment . . -273 

344. Subjective character of Syllogism and Induction . . .276 

345. Terms antithetic to ' Subjective' and 6 Formal' . . . 279 

CHAPTER V. 

THE A PRIORI TRUTHS. 

346. The world of Knowledge and the world of Things . . 283 

347. ' Actual Reality ' ; adequacy of Judgments to it . . . 286 

348. Applicability of thought to the course of events involves 

(1) Some given reality, which thought cannot create . 288 

349. (2) The Universality of Law in the Real world ; ultimately 

a matter of faith 290 

350. And (3) synthetic judgments a priori, as basis of knowledge 

of particular laws 294 

351. Hume's restriction of judgment destroys all judgment . . 295 

352. Mathematical reasoning is not covered by the Law of Identity 297 

353. Illustration by Kant's arithmetical instance . . . -299 

354. And by his geometrical instance 303 

355. Meaning and value of apprehension a priori . . . 305 

356. Self-evidence of universal Truths 307 

357. Intuition is opposed to discursive thought — means immediate 

apprehension 309 

358. Self-evident Truths require to be discovered by help of 

analysis 311 

359. Pure Mechanics in what sense a priori . . . . 313 

360. Gradual formation of pure ideas of Motion and Mass . . 316 

361. Mechanical principles, like those of Arithmetic and Geometry, 

at once identical and synthetic 319 



xii 



TABLE OF CONTENTS. 



PAGE 



362. In higher Mechanics, Proof is one thing, and the Ratio legis 

another 323 

363. Analytical Knowledge as the ideal, means the simplest syn- 

thetical knowledge . . . . . . . -325 

364. The simplest ultimate Truth need not be a mere datum of 

experience, though it must be Synthetic . . . . 327 

365. A synthetic yet necessary development the supreme goal of 

science . . . . . . . . . . 329 

Appendix . . 331 

Index ... . . . ... . . . . -333 



CHAPTER VI. 



Fallacies and Dilemmas, 

240. True conclusions, as Aristotle has observed, can be 
correctly drawn from false premises. Every Laplander is 
a born poet, Homer was a Laplander, and therefore — by 
the first figure — a poet. All parasitic plants have red 
flowers, no rose has red flowers, therefore — by the second 
figure — roses are not parasitic plants. Metals do not con- 
duct electricity, all metals are non-fusible, and hence — 
according to the third figure — non-fusible substances exist, 
which are non-conductors of electricity. Alter Laplander 
into Greek, plants which have red flowers into plants which 
have exploding seed-vessels, and write glass for metal, and 
in each example one premise will be true, while by inserting 
a new middle term in each case you may make both pre- 
mises true, but in every case the conclusion follows with 
neither more nor less validity. Let T be a perfectly true 
proposition, S its subject, and P its predicate ; then a 
middle term M may be chosen at random so long as the 
terms are arranged in both premises on the model of an 
Aristotelian figure : if this is done the conclusion T will 
always follow according to the figure. 

We shall see why this is universally true, if we take as 
our middle term an abstract symbol Af, instead of a con- 
crete term : thus, all M are poets, Homer was an M; all 
parasitic plants are M, roses are not M ; all M are non- 
conductors, all M are non-fusible. What these symbolic 

Logic, Vol. II. B 



2 



FALLACIES AND DILEMMAS, 



[Book II. 



premises tell us is the relations in which S and P must 
stand to some middle term, if their conjunction SP is to be 
valid in the conclusion : and conversely these premises tell 
us that given any middle term M to which *S* and P are 
related as required, then the proposition SP must be valid. 
If the M is found and so both the required premises 
established, then SP is valid not merely in fact but now 
also of necessity ; on the other hand if we could show that 
there exists no M to which S and P can stand in the 
requisite relation, we shall know that S P was impossible, 
for no experience could give us £ P as a fact : but if we 
have merely chosen a wrong M then the case is different. 
The premises we have chosen will not do, but that is no 
reason why there should not be some other M, the insertion 
of which will render the premises correct and so necessitate 
the conclusion SP. If again we have correctly drawn a 
conclusion SP and that conclusion is unsound, there must 
be something false in the premises, from which it follows. 
In a word in all cases where T is not given in direct per- 
ception, but deduced from premises, what really depends 
on the correctness of those premises is not the truth of T, 
but only our insight into that truth. Without correct pre- 
mises T cannot indeed be proved, but nevertheless it can 
be true and its truth is independent of any errors we may 
commit, when reflecting about it, and subsists even when 
conclusively deduced from premises materially false. This 
point deserves notice, for it is a common mistake in rea- 
soning to take the invalidity of the proof which is offered 
for T as a proof of the falsehood of T itself, and to confuse 
the refutation of an argument with the disproof of a fact. 

241. A proposition T is valid if it is rightly drawn from 
valid premises, but it is not proved unless these premises 
are valid independently of itself. If T itself or any propo- 
sition T\ whose validity presupposes the validity of T, 
appears disguised in the premises, T is correctly deduced, 
but is not proved at all. This fallacy is called petiiio prin- 



Chap, vi.] HYSTERON PROTERON. 



3 



cipii or circulus in denionstrando, and in its naked form seems 
easily avoided. Yet it is frequently committed, especially 
where the conclusion is reached by a long chain of de- 
duction and depends on the constructions of the scientific 
imagination as well as on the relations of abstract ideas. 
In such a case we are often able to deduce T with formal 
accuracy by first presupposing some indirect and distant 
consequence of T, which consequence of T is then taken 
as an independent truth from which T follows. There are 
no rules which will enable us to avoid this mistake, but it 
may be well to remember that we are peculiarly liable to it 
when we attempt to prove by a direct and progressive 
argument propositions which contain some final and un- 
derivative element of our knowledge. In such cases, 
whether the element be a necessity of our thought or a 
fact universally valid in our perception, apagogic and re- 
gressive methods alone are applicable. 

242. The second kind of fallacy is called Hysteron Pro- 
teron. It is so like the first (the argument in a circle) that 
we often have no reason to distinguish it therefrom. It 
consists in using a proposition, which both calls for and 
admits of proof, to demonstrate another, which not only 
needs none, but is itself actually the proper ground from 
which to prove the first proposition. We are told for 
example that God's will is holy, that the moral dictates of 
our conscience are the expression of the divine will within 
us, and therefore they too are holy and binding on us. But 
we cannot help objecting that if the holiness and binding 
force of our moral dictates were not felt by us as an inde- 
pendent fact and irrespectively of the origin of those dictates, 
the argument would fall to the ground. Upon other grounds 
no doubt we might continue to believe in a mere supreme 
being, but the idea of holiness would not and could not 
suggest itself to us, and hence the major premise of the 
argument proposed could never exist. The transition from 
God's will to our conscience is therefore no proof ; but 

B 2 



4 



FALLACIES AND DILEMMAS. [Book II. 



although inadmissible as a sequence of thought it is perhaps 
the right way of giving expression to the truth. For in a 
great many cases that which is in fact the consequence or 
principiatum may be for us a means, and often the only 
means, of knowing that which in itself is the principium or 
real ground of the possibility of the former. When we have 
acquired knowledge, by way of induction especially, and are 
exhibiting the result systematically, it is evident that we 
always take the universal statement, which we really know 
only from the particulars, and placing it at the head of them 
use it to prove those very particulars. Hence it is of im- 
portance alongside of this method to employ another mode 
of exposition which shall set out the items of our knowledge 
in the order in which they can actually be proved by the 
help one of another. We often allow ourselves to commit 
a hysteron proteron, when we are trying to prove a point, 
either in the course of conversation, or in the rapid reflexion 
by which we seek to assure ourselves of the truth of some 
proposition which we desire to employ in an enquiry. The 
inference in these cases is ex concessis, from premises whose 
truth we presuppose but do not discuss. In an enquiry the 
implication of these premises with the rest of our knowledge 
is taken as a sufficient guarantee, and in conversation we 
may find it easier to get these premises admitted than it 
would be to gain acceptance for the truth from which in 
reality they follow. 

243. The commonest fallacy is ambiguity of the middle 
term, quaternio terminorum or fallacia falsi ?7iedii more or 
less disguised. The Greek sophists were the first to remark 
the chain of thought which appears in the syllogism and to 
notice its linguistic expression, and a great number of these 
fallacies were at that time exhibited. They are classified 
in the Aristotelian work on the subject, but many of them 
have no value at the present time, even in the light of 
pleasantries. There are yet some which remain as abiding 
sources of danger, and among these we may signalise the 



Chap. VI.] DE DICTO— AD DICTUM—, 



5 



double fallacy de dicto simpliciter ad dictum secundum quid 
and de dicto secundum quid ad dictum simpliciter. Two 
general modes of fallacious thought are developed by the 
habitual commission of these fallacies and illustrate them on 
a grand scale. The first is doctrinairism, the second 
narrow-mindedness. The doctrinaire is an idealist, who 
refuses to see that though ideas may be right in the abstract, 
yet the nature of the circumstances under which and of the 
objects to which they are to be applied must limit not only 
their practicability but even their binding force. The 
narrow-minded, on the other hand, can recognise and 
esteem no truth and no ideal, even the most universally 
valid, except in that special form to which they have 
become accustomed within a limited circle of thought and 
personal observation. Life is a school, which corrects these 
habits of mind. The parochially-minded man sees things 
persist in spite of himself in taking shapes, which he con- 
siders unprecedented, but he finds the world somehow 
survives it and learns at last that a system of life may be 
excellent and precious, but that it is rash from that to argue 
that it is the only proper mode of orderly existence. And 
the enthusiast for ideals, when he sees the curtailment 
which every attempt at realisation inflicts on them, learns 
the lesson which the disjunctive theorem might have taught 
him. Every universal P changes in the act of being applied 
from something that held simpliciter into something that 
holds secundum quid, — changes from P to p 1 p 2 or p 3 : to 
refuse to accept it in any one of these, which are its only 
possible shapes, is to ask that it be realised under a condi 
tion which even logic pronounces impossible. 

244. One of these fallacies consists in our taking a P, 
which holds good of M in the abstract, and asserting it of 
M under new conditions which make it no longer applicable. 
The other and opposite mistake transfers to M taken abso- 
lutely predicates, which are only true of M under certain 
conditions. In both we have an ambiguity of the middle 



6 



FALLACIES AND DILEMMAS. 



[Book II. 



term, which wavers in its meaning between the unlimited 
M and the determinate conditioned M\ Examples present 
themselves and could be given in great numbers, but there 
is one worth dwelling on from its own special interest. 
I refer to the question of the morality of lying. On principle 
we condemn all lies, but in practice almost everyone allows 
there are exceptions, a confession which points to some 
mistake in the way in which the principle is laid down. In 
fact the particular lies which, apart from the influence of 
education, we find hateful are those whose object is to make 
others chargeable for our faults, wantonly to do mischief, or 
to wound the self-esteem of another by entangling him in 
false ideas in order to exalt ourselves. It is these secondary 
features that rouse our indignation against an untruth, and 
it is only these that make us call an untruth a lie. The 
secundum quid and its influence on our judgment is quite 
plain in these cases, and on the strength of them apart from 
other considerations we should be wrong to conclude that 
every falsehood, when the intention is not bad, is immoral. 
Other considerations do however exist. We communicate 
with our fellows in order to waken in all alike ideas of the 
same reality, and our object in doing this is that when we 
work together our efforts may coincide, and when we work 
apart we may avoid collisions, and in general we desire to 
avoid undertakings which are not in accordance with the 
nature of things. But failure would be certain, if everyone 
made it the rule to lie ; everywhere the truth is one, but 
possible falsehoods are innumerable. The interchange of 
falsehoods therefore ensures no meeting-point for common 
action, so that however good our intentions we should ever 
be missing the mark. False assertion is thus contrary to 
the essence of assertion, to the moral end which all com- 
munication aims at, and therefore we set it down broadly as 
a thing in itself reprehensible. The untruths of poetry, 
jesting and courtesy are exceptions, they are not real asser- 
tions, and on these points we make a silent reservation. 



Chap. VI.] 



EXCEPTIONAL CASES. 



1 



Here comes in the fallacy. We think that we can now un- 
conditionally assert the badness of falsehood ; that we have 
got rid of the old secundum quid and have got the simpliciter. 
Unfortunately the simpliciter is ambiguous ; it may mean 
that false-speaking is wrong in itself 'and can be justified 
only secundum quid, i.e. for special reasons in particular 
cases. But it may mean that falsehood is universally bad, 
so that no special considerations can ever justify it. These 
versions of the simpliciter practically collide in our con- 
sciences, and it is that which makes our opinions about 
falsehood so self-contradictory. The logical premises from 
which we here started justify only the first version. What 
we started from was that universal false-speaking would 
frustrate our moral aims and therefore we said it was bad ; 
but where the aim is immoral it may be right to frustrate it, 
and admitting our premise it is therefore still possible that 
a lie may be justified. To prove the simpliciter in the sense 
of without exception other premises would be wanted. It 
would be the business of ethics to discuss them, we are 
here concerned with only the logical side of the question, 
and our object has been to show that fallacia falsi medii 
arises not only through confusion of simpliciter and secundum 
quid, but also that the simpliciter itself in the example we 
discussed and in many others beside is the seat of an 
ambiguity. A thing true simpliciter may be true by itself 
alone and not under all conditions, i. e. it may be true only 
in general, but not always and in all particular cases. But 
it may equally well be true by itself "in the sense of being 
true independently of conditions and hence always and 
necessarily true in every particular case ; universally, that 
is, not merely in general. 

245. We may consider here some examples, where an 
universal proposition is extended to instances which can 
formally be brought under it as exceptional cases, but 
where the conditions which make it applicable have dis- 
appeared. If the terms of the proposition are variable 



8 



FALLACIES AND DILEMMAS, [Book II. 



quantities, and if these are followed to their limits at 
zero or infinity, we get such examples. With a lever 
the work done remains the same, so long as the product 
w I of the weight w into the arm / is unaltered. Thus 
the more / is increased, the less weight w is wanted to 
do the same work; hence, it has been subtly argued, at 
an infinite distance from the fulcrum, a mass = o would 
suffice to balance any weight whatever on the other arm 
of the lever ; and this conclusion has been urged against 
the validity of the general formula. It is natural simply 
to dismiss the idea by remarking that the formula con- 
templates cases where real forces are applied to the lever, 
and is not true where that condition is wanting. This 
removes our doubts on the question of fact, but hardly 
settles the logical problem. For we do not always dismiss 
these cases ; we have no hesitation in taking cos o — i, 
although the idea of a cosine is in its origin without 
meaning except for a real arc <£, from the extremity of 
which a perpendicular may be drawn to the semi-diameter 
through the initial point, and we pass from this case to 
the limiting value <£ = o. Now since the law of the 
lever remains valid at every stage of approximation to 
the values / = oo and w — o, it would be well if it admitted 
of being interpreted for these limiting cases in such a 
way as to show what is the second meaning which it 
assumes after the first becomes inadmissible, or failing 
this it would be desirable that the formula itself should 
exhibit its own invalidity. This it would do not merely 
by producing conclusions, which from a point of view 
external to the formula we can judge to be incredible, 
but still more by becoming destructive of itself. The 
force which a wedge exerts varies inversely with the breadth 
of its back ; let this sink to nothing and we get the same 
dilemma : the formula gives an infinite effect, while the 
effect is in fact nothing at all. But here we might answer, 
though more in jest than in earnest, that as a matter of 



Chap. VI.] 



THE LEVER. 



9 



fact it would need an infinite force to keep a geometrical 
plane, such as we have now reduced our wedge to, from 
penetrating a block of wood ; and it might be proved 
with equal show of . formal rectitude that this would not 
require the block to be cleft. 

I cannot at present as decisively settle the doubts which 
some have about the lever; though I should deem it 
irrational to postpone any consideration of the principle 
of the lever till one had solved the problem which arises 
in connexion with an arm of infinite length, for surely it 
is intrinsically absurd to think of the mass = o at an infinite 
distance as having any effect at all; the very idea, as it 
admits of no interpretation in fact, must be self-contra- 
dictory. And so it is, for the law has no meaning except 
as ascribing to a definite mass w at a definite distance / 
from the fulcrum a definite effect, which alters as / alters. 
Now why should a mass == o produce an effect at an infinite 
distance other than it would produce at any finite distance 
we like to take ? How would the case of a mass = o 
acting at the end of a lever of infinite length differ from 
that of a mass = o acting at any other point in the lever, 
or from a third case, which would properly always coexist 
with the other two, viz. that in which we suppose the 
nothing to be applied at all points in the lever and — 
what is more — to act in any direction we like? Thus 
the attempt to retain the law of the lever for zul—o .00 fails, 
not merely because it gives incredible results, but because 
the law loses all its meaning inasmuch as things become 
indistinguishable whose distinction is essential to that 
meaning. Other paths lead to the same conclusion, wl 
is no constant quantity, so that w should vary inversely 
as /, but the effect varies for every and any / with the 
variation of w, and w is quite free to vary as it will 
consequently the effect becomes = o, when w sinks to o, 
no matter what / may be ; it follows that wl = o . 00 can 
only have the value o and no other. 



io FALLACIES AND DILEMMAS. [Book II. 

246. There is another fallacy which is akin to that of 
too wide or too narrow definition and in general arises 
from it. T being the point to be proved, the mistake 
consists in proving too much or too little either as to 
the qualities which T includes, or the extent to which 
it is applicable. The conclusion which proves too much 
may be correct and may err simply in going further than 
was required. For instance you may prove correctly of 
all animals what you wanted to prove only of men, and 
in such a case the ground is valid and has simply been 
applied more generally than was requisite. But in other 
cases a conclusion may be false as well as too wide, and 
here the mistake lies in using a wrong ground of proof, 
so throwing doubt upon the narrower conclusion instead 
of proving it. In the argument which proves too little, 
the mistake is again of two kinds. In the first case the 
ground of inference may be a true and general proposition 
quite wide enough to prove T, and the mistake lies in 
taking this ground in a particular form which will not do 
so ; it is corrected simply by using the ground of inference 
in the general form in which it proves T. But the second 
form of the fallacy is more serious. A special case of T 
may have been correctly proved from certain premises, 
but those premises may be quite inadequate to establish 
jTas an universal. To sum up, any argument which does 
not exactly tally with the proposition to be proved must 
leave us in doubt as to its applicability; nihil probat 
qui nimium probat is as true as nihil probat qui parum 
probat. 

247. It would be easy to supply illustrations, but I 
prefer to consider another fallacy, that of incomplete ex- 
planation. This is often to be met with in the speculations 
of amateurs, but does not generally take the form of 
demonstration. It consists in assigning a general cause 
for some phenomenon without enquiring if the cause as- 
signed will account for the particular modifications, to 



Chap. VI.] 



A TOO GENERAL CAUSE. 



1 1 



which that phenomenon is subject. It is perhaps not 
possible to deduce the law of the persistence of motion 
from any more universal principle; but at any rate the 
vulgar opinion that every motion ceases with the lapse 
of time is impossible in itself, and can be used to prove 
the law by a reductio ad absurdum. That diminution of 
motion arises from real obstacles existing in time is true 
and is quite conceivable, but it is quite inconceivable how 
mere time should cause a diminution. No doubt our own 
bodily movements are enfeebled as exhaustion increases, 
and this might suggest to us the idea that mere lapse of 
time can destroy all motion; yet we are undeceived when 
we reflect that if this really happens it must happen either 
earlier or later, at some one particular time or another, but 
that there exists no law and no reason to connect it with 
any particular time. For assume that each of the ab- 
solutely similar moments dt has the same constant con- 
suming power, and takes from every unit of mass the 
velocity q, — no doubt one can understand on this as- 
sumption, how it is that swifter motions of the same mass 
persist longer than slower motions. But on the other 
hand so long as q is a finite quantity we can still think 
of motions, whose velocity during dt regarded as one 
period is less than q, and these movements could on such 
an assumption never take place in reality at all. Or shall 
we assume that the consuming force of time is proportional 
to the velocity to be consumed? Then the question 
remains as to the ratio. But I refrain from any further 
hypotheses. In the first place since time and mass are 
utterly disparate things one sees at once how hopeless it 
is to try to determine a unit of mass, for which q would 
measure the arresting power of a time d t. In the second 
place we can equally conceive of innumerable different 
ratios as existing between the velocity and rate of re- 
tardation, and it is evident that mere time can of itself 
afford us no reason for preferring one of these ratios to 



1 2 



FALLACIES AND DILEMMAS. 



[Book II. 



another. But apart from all this there is an objection 
which would render all such attempts idle. For supposing 
that a time d t removes some part of the motion, the 
question remains — whence comes the residue, the motion 
which has not been removed? It is clear that in case 
of this residue the law of the persistency of motion has 
been presupposed. If we had not tacitly presupposed 
the law to be valid for it, we should have to say that the 
whole motion was arrested by the first period d t. What 
it comes to then is this. Either motion does not take 
place at all but vanishes 1 the moment it gives a sign of 
intending to endure through a time d t ; or if motion 
gradually diminishes, the law of persistence is the primary 
law, and the diminution of motion is secondary, the result 
of external obstacles. These obstacles we shall now natur- 
ally seek only in what is homogeneous with itself, viz. 
in opposed tendencies to motion. 

I will merely call the reader's attention to the connexion 
of this proposition (that every ground of explanation must 
establish not merely T in general, but also the possibility 
of its modifications) with the doctrine of the disjunctive 
judgment. It would take us too far afield into mathematics, 
if we followed up this clue ; it is sufficient just to notice 
that this logical requirement has found a special and 
fruitful expression of itself in the principle of the homo- 
geneity of the functions to be combined in an equation. 

248. Incomplete disjunction is a fallacy which often occurs 
in collective and indirect arguments. In order to prove T, 
these arguments attempt either to show that that holds in 
all particular cases of T, which it is desired to affirm 
universally, or to establish T as the sole remaining pos- 
sibility by first disproving all cases of non-Z Neither 
task is very easy. In practical life especially we often 
find it very difficult in laying down a rule to examine 
beforehand all the possible cases to which it may be 

1 [Compare Metaph. § 163.] 



Chap. VI.] INCOMPLETE DISJUNCTION. 13 

applied and to see whether the proposed regulation would 
always be desirable or tolerable; and it is common again 
to find that after considering every course which seems 
conceivable, and after concluding all but one to be im- 
possible, a momentary inspiration will suggest some other 
way out of the difficulty, which we had quite overlooked. 
In theory the most fruitful source of the fallacy is the 
dominant influence of some one order of ideas. Instead 
of setting out all the possible alternatives we are led silently 
to the one which consorts best with our own one-sidedness. 
For instance our sensations are subjective states excited 
in us; it is easy to show this; and further according to 
a view now widely accepted the forms of space and time 
in which we arrange the manifold matter of sensation 
are subjective also, they are modes in which our minds 
perceive. From this point we are easily led on to regard 
the idea of unknown things and processes underlying phe- 
nomena as a creation of our mind, which is compelled 
by its constitution to adopt this means of giving unity 
to its singular perceptions. The subjectivity of all elements 
of knowledge is thus established, and finally we venture 
on the inference that therefore there is no objective real 
world answering to the world of our ideas. But the in- 
ference is false : for supposing that this real world does 
exist, it is easy to see that things would be just the 
same to us as we find them. If real elements exist they 
can never pass into us bodily : they can do nothing but 
excite in us sensations and ideas, and these though caused 
by external impressions and our reaction against them 
would still be a creation of our subjective nature. An 
objective space may exist or may not exist, but at all 
events the perception of it must be the product of our 
subjective faculties. So too with causality : whether the 
law has objective validity or no, in either case equally 
we are forced to recognise it only so far as we think it 
and perceive its content to be in harmony with the laws 



FALLACIES AND DILEMMAS. [Book II. 



of our thinking. Thus we see the complete subjectivity 
of all the elements of our knowledge proves nothing what- 
ever as to the existence or non-existence of an objective 
reality. The best preservative against this kind of fallacy 
is the existence of hostile opinion. It is indispensable, 
besides developing our own doctrine, to familiarise ourselves 
with ideas which proceed from points of view opposed to 
our own. 

249. Sophisms are distinguished from paralogisms. The 
latter are involuntary mistakes in inference, while the former 
are intentionally fallacious arguments, whose object is to 
confuse or deceive. It is thus obvious that in cases where 
the intention is doubtful we cannot tell a paralogism from a 
sophism. Zeno's arguments against the reality of motion 
may be taken as either. I shall not here discuss those real 
difficulties in the idea of motion which they touch upon, 
but will exhibit the arguments as examples of fallacies that 
are difficult to classify. The first argument tries to prove 
that an arrow in flight is at rest. It starts with the assump- 
tion that time consists of indivisible moments, and then 
infers that the arrow cannot move in any one of these 
moments. For if it moves, it must now be at one place 
and then at another, but in each indivisible moment there 
is no now and then, no before or after. Therefore the arrow 
is at rest in each, and if in each then in all ; hence it is 
always at rest. To this we may answer simply that rest also 
is impossible unless a thing is at one and the same place 
both now and then, both before and after ; and since in an 
indivisible moment there is no before or after the arrow can 
no more rest in it than move. And this conclusion accords 
with the ordinary theory of motion. So long as dt is a 
length of time the arrow passes in it through a small space 
v. d t, but as soon as ever d t ceases to be a quantity and 
becomes a mere dividing point in time with its definite 
position in the series of time — then no doubt in dt the 
arrow passes through no space at all ; still it does not rest 



Chap. VI.] 



ACHILLES AND THE SNAIL. 



15 



in it, but goes through it with a velocity v. But apart from 
this objection, what right has Zeno to maintain that in each 
successive moment of the arrow's flight it is at rest in the 
same place as it was at rest in before ? There is nothing, 
I admit, in the idea of a moment or an arrow as such from 
which the idea of a change of place could follow. But it 
surely is involved in an arrow's flight. No doubt there is a 
difficulty as to the nature of that impulse which makes a 
body in motion different 1 at every moment from a stationary 
body, even if the moment be conceived as indivisible. And 
this is the point against which a sophism might have been 
directed with effect. But, failing this and failing any proof 
that velocity could not exist, Zeno had no right to start with 
the flying of an arrow, — so assuming velocity, — and then in 
his proof to drop the idea out of sight. All his argument 
shows, if you take it as it stands, is that rest is not motion 
and that motion can never be compounded out of rest. 
Had he retained the idea of velocity Zeno could at least 
have deduced such a successive change of place as pro- 
ceeded by jumps from moment to moment ; the conception 
of continuous motion he could not get at so long as he 
held by the notion that time is made up of indivisible 
moments. 

Another of his arguments was that if the snail has a start 
the swift-footed Achilles can never catch it up, inasmuch as 
before ever he can overtake it he must first reach the place 
it has just left before and so on for ever. The argument 
might be simplified by omitting the fact of the snail's 
movement, which for the particular paradox is superfluous. 
Did the snail remain still Achilles would yet never reach it. 
Nothing which is in motion — this is the real basis of the 
argument — can ever come to the end of any given space 
however small. To do so it must first accomplish half the 
given distance, then half the remainder, then half the re- 
maining quarter and so on for ever, so as never to reach the 
1 [Compare Metaphysic, § 171.] 



1 6 FALLACIES AND DILEMMAS. [Book II. 

end of the space. The argument assumes that the halving 
process can go on for ever ad infinitum, and so presupposes 
that the space is infinitely divisible or already consists of an 
infinite number of parts. It further assumes that an indivi- 
sible moment of time is required for the transition from one 
point of space to another ; and so concludes that an infinite 
number of moments must elapse before any space whatever 
is got over, since any space contains an infinite number of 
parts. The conclusion, if we admit the premises, is so far 
quite correct ; but it is quite an arbitrary assertion that this 
aggregate of infinitely numerous moments would form an 
infinite length of time ; seeing that they are indivisible each 
of them must contain no sooner or later, unless, as Zeno 
does here, we foist in between every two moments a real 
lapse of time or represent these moments as following each 
other at definite intervals in a sort of secondary time, which 
we imagine as filling up the background. It is not even 
necessary to object to Zeno, what Aristotle's remark on the 
subject amounts to, that (according to our modes of ex- 
pression) the integral of an infinite series of quantities 
continuously passing into one another may still be a definite 
finite quantity, and that therefore the aggregate of moments 
of time may be a finite length of time. The indivisible 
moments of time are conceived by Zeno not only as being 
each for itself without quantity, but also as so detached 
from each other, that there is no question of a transition, in 
virtue of which they would become constituents of time 
at all. The sum of all these nothings is therefore itself 
nothing ; it is only an unwarranted complaisance on the 
part of our better-instructed fancy which allows it to be 
passed off on us as a quantity at all and then as an infinite 
quantity. Achilles requires no doubt an infinite number of 
moments of time to get from a to but these do not make 
up any length of time at all ; it would be truer to say that 
Achilles consumes no time at all than that he consumes an 
infinite time ; indeed it remains hard to say what end is 



Chap. VI.] 



PSEUDOMENOS. 



17 



served in this connexion by the consumption of indivisible 
moments of time or what it means. 

250. Besides these fallacies the ancients have handed 
down to us many interesting dilemmas, i.e. conjunctions 
of thoughts from which follow opposite conclusions, equally 
necessary and equally impossible. One dilemma nick- 
named Pseudomenos dates from Epimenides, who being a 
Cretan himself asserted that every Cretan lies as soon as he 
opens his lips. If what he asserted was true, he himself 
lied, in which case what he said must have been false ; but 
if it is false it is still possible that the Cretans do not always 
lie but lie sometimes, and that Epimenides himself actually 
lied on this occasion in making the universal assertion. In 
this case there will be no incongruity between the fact 
asserted and the fact that it is asserted, and a way out of the 
dilemma is left open to us. Not so however if we drop 
Epimenides and the Cretans and instead of these two 
subjects, one of which is only contained in the other but 
not identical with it, put an identical subject : c I lie now.' 
If my assertion is true, i.e. if I am lying, what I assert is 
false, and I am not lying. But what I assert is that my 
assertion is untrue : if it is false to say that my assertion is 
untrue, my assertion becomes true again and I am lying, 
and the whole chain of self-destructive consequences begins 
afresh. The reason of them is easily detected. Logically 
of course what is asserted is true or false, quite apart from 
the fact of its being asserted ; it may be asserted or it 
may not : but the only sort of truth or untruth which the 
assertion can have, is what it acquires through the truth or 
untruth of what is asserted, which is independent of it. 
Thus we get contradictions, some of which are formally 
insoluble, when what is asserted is such as to involve some- 
thing in regard to the fact of its assertion which makes the 
assertion impossible or untrue. The difficulty vanishes if 
instead of saying: I lie, we say: I lied. Just as little can 
we say in the present tense, I am silent, though we can 

Logic, Vol. II. C 



FALLACIES AND DILEMMAS. 



[Book II. 



quite well use the future, I will be silent, for then our 
assertion refers to another fact than itself, to a fact which is 
not in conflict with it. 

There are many other instances of the sort, though none 
so classical as Pseudomenos. If a person answers yes, 
when he is asked if he he asleep, he sets his assertion in 
the same sort of conflict with what he asserts ; so does a 
person, who calls out to an unwelcome visitor, that he is 
not at home. Lastly there are other cases resembling 
these, cases in which one subject has in an impossible 
manner to form both terms at once of a relation, which 
can only exist between two different subjects : Jean Paul's 
dwarf for instance, who only reached up to his own knees, 
not to mention other people's ; or the inscription over the 
elephant's booth : this is the biggest elephant in the world, 
itself excepted; or lastly we may instance Munchausen's 
kind service to himself, when he pulled himself out of the 
bog by his own hair. Equally curious is the old dilemma 
of the crocodile : quoth the brute to the wailing mother : I 
give you back your child, if you tell me truly whether I am 
going to give it back or no. It would all be plain-sailing 
had the mother only to guess, if the crocodile intends at that 
moment to give or not give it back. If she guesses right, 
there is nothing to prevent the child being restored accord- 
ing to agreement ; for even were it true, that the crocodile 
does not just then intend to restore it, still, if her guess is 
that he does not intend, he may yet fulfil the contract by 
changing his intention or giving it up against his will. But 
if she guesses wrong, she loses all claim to have her child 
back again. For whatever may have been the animal's real 
intention — which she guessed wrong — he need not in his 
actions bind himself by his then intention ; he need only 
observe the terms of the contract, and this now that she has 
guessed wrongly forbids him to return the child. But the 
question asked of the mother, whether she is going to get 
her child back or no, need not refer to the animal's inten- 



Chap. VI.] CLASSICAL DILEMMAS. 19 

tions ; we may conceive of this future as a predestined to- 
come, so that in itself it is already a settled matter, which of 
the two possible events is going to occur. Interpreting the 
question in this way, we get an insoluble perplexity for 
obvious reasons ; for we cannot without absurdity make an 
issue, which unconditionally impends, depend on a con- 
dition, whose fulfilment would necessarily be as ineffectual 
to bring about — as its non-fulfilment to bar — the inevitable. 
There is thus but one way out of the dilemma. If the re- 
storation of the child is the event which is going to occur, 
and if the mother guesses this, all will end happily, yet not 
because her assertion in any way conditioned the happy 
result : her assertion in itself is quite ineffectual ; it has 
only chanced to agree with the inevitable result and the 
terms of the bargain. If she had made any other answer, 
that would only have served to reveal more plainly the 
utter impotence of a bargain, which because it tries to con- 
dition the unconditioned must of necessity be violated. 
But the old form of the dilemma starts from yet another 
supposition, different from both of these : it supposes that 
it is not determined beforehand which of the two alternatives 
is going to occur, i. e. whether the child is going to be given 
back or not ; that is to be settled by what the mother says. 
Now logic teaches that in any hypothetical judgment the 
validity of the consequent rests on that of the antecedent ; 
but the latter must be independently fixed and unambiguous 
and must neither in its meaning nor its validity be con- 
ditioned by the meaning and validity of the consequent. 
In the case before us this absolute requirement is contra- 
vened. For the condition fixed upon here is not an asser- 
tion made by the mother but the truth of her assertion, and 
further not the truth of an assertion which refers to some 
third fact independent of the future result and which could 
therefore be true or untrue no matter what this result may 
be ; on the contrary what she asserts is that this result will 
either occur or not occur, — a result which is connected with 

c 2 



20 



FALLACIES AND DILEMMAS. [Book II. 



no other conditions at all, — and so its own truth depends on 
the very thing, which should depend on it. Consequently 
on this supposition as on the former there is only one case 
which logically admits of a satisfactory issue; the mother 
must answer, you are going to restore my child \ and then 
its actual restoration at once makes her answer true and 
fulfils the agreement. In that case the issue is a happy one, 
but it was not conditioned by the answer she gave. Suppose 
after all the crocodile does not give back the baby, the very 
fact of his not doing so makes her answer a false one and at 
the same time the animal becomes justified according to 
the terms of the agreement in not giving it back. If how- 
ever the mother is so unfortunate as to answer : you will 
not give it back, 6 then ' the crocodile must say ' I cannot 
give it back ; the agreement forbids me to, since if I did 
your answer would become a false one ; no more could I 
restore the child even if your answer could be correct, 
seeing that by the very fact of my returning it it would be- 
come false.' The mother then objects : ' you must in any 
case give it me ; on the score of the agreement, if my answer 
was correct ; but no less if it was incorrect ; for it would 
become a correct answer, if you refused to give it back.' 
There is no way out of this dilemma ; as a matter of fact 
however both parties rest their cases on unthinkable grounds ; 
for the answer really given can as little be true or untrue in- 
dependently of the actual result as could the answer she 
might have given, an answer which only differs from this in 
being more fortunate. 

The dilemma of Protagoras and Euathlus rests on a similar 
misuse of hypothetical conjunctions of thoughts. Euathlus 
is to pay for the instruction he has received as soon as he 
wins his first case ; but as he engages in no suits, Protagoras 
gets nothing and sues him on that account. Now whether 
Euathlus wins or loses this suit, the verdict must in any case 
either oblige him to do that which the contract releases him 
from doing or release him from doing that which the con- 



Chap. VI.] EUATHLUS AND PROTAGORAS, 21 

tract obliges him to do. Various solutions of the difficulty 
have been attempted on the supposition that Euathlus is 
allowed to win this his first suit because he has won no pre- 
vious suit, and so had not yet become obliged to pay. It 
was then open to Protagoras to institute a fresh suit, which 
must have this time led to his pupil being condemned to pay. 
This would be shifting an absurdity off logic, in order to 
make a present of it to jurisprudence. I will not anticipate 
the decision of the latter, but I suspect it would say that in 
acting as he did Euathlus had fraudulently prevented a 
certain condition from being realised, according to which 
he would have been forced to fulfil an obligation. If there- 
fore it could fix a date, after which no other interpretation 
could be put on his conduct than that it was fraudulent, 
then though Protagoras no doubt could not base a suit on 
the contract, the law might well go behind it and taking its 
stand on the obligation, under which Euathlus really put 
himself by receiving the instruction, condemn him to pay, 
just as if the ambiguous agreement had never existed. 



CHAPTER VII. 

Universal propositions as derived from perceptions. 

251. The ideas which we ourselves have put together are 
completely open to our inspection and we can review their 
content and manner of conjunction. And hence the con- 
clusions we draw from them are necessary and the process 
of conclusion is proof or demonstration, the essence of 
which is to descend from the more to the less universal and 
starting from a general truth to end with a particular appli- 
cation thereof. But the conjunctions of phenomena in the 
world outside us do not carry on their faces the universal 
laws and conditions of their connexion. They are individual 
experiences to be severally expressed in particular propo- 
sitions, and though each embodies an universal principle, 
yet the path up to that principle must be a matter of search. 
The simplest form of this ascent in thought is a process 
with which we have become familiar as the inductive syllo- 
gism, and hence it is the custom in our day to collect into 
one body the numerous operations which assist us in 
ascending to generals and to call this inductive logic and to 
set it against the deductive or demonstrative logic along 
with much disparagement of the latter. Such disparage- 
ment rests on a mistake. The inductive methods it is 
certain are the most effectual helps to the attainment of 
new truth, but it is no less certain that they rest entirely on 
the results of deductive logic. It is the theory of the validity 



PURE CASES. 



23 



of syllogisms, the convertibility and contraposition of judg- 
ments, and of the forms of proof, that is the source of each 
provision and precaution by which so far as may be we 
secure each step of our paths as we ascend by induction 
from given perceptions to the universal laws of the real 
world. 

252. The first step of this ascent is barred we are told by 
an insuperable obstacle. Experience we are told cannot 
give us universal cognitions, and in one sense no doubt this 
well-worn saying is true : but if we take it to assert a differ- 
ence in validity between two sources of knowledge, experi- 
ence on the one hand and an a priori certitude on the other, 
then the saying is true no longer and is the opposite of truth. 
Every experience, whose contents in their connexion can be 
expressed without deficit or surplusage in the form S is P, 
must ipso facto rank as an universal judgment, even if this 
experience stands absolutely alone. The law of identity 
guarantees that if the same S were once more perceived in 
a second experience it would be impossible that the same 
predicate P should be absent or should be replaced by some 
other predicate Q. On the other hand it is no less true 
that experience does not directly present us with percep- 
tions which fulfil this condition. Our perceptions do not 
give us a subject -S 1 in conjunction with a predicate P and 
nothing more or less than this subject and this predicate. 
The real and true subject with which the phenomenon we 
observe is essentially connected is not S but 2. It has ele- 
ments s absolutely necessary to the production of the pheno- 
menon and which notwithstanding we do not perceive. 
What we do observe, S, is a residue and what is more an 
impure residue, for it comes to us indissolubly joined with 
elements 0-, which have nothing whatever to do with the 
production of the phenomenon. It is the same with the 
predicate. The true predicate which attaches to 2), the 
true subject, is II and it we do not perceive. It has features 
/ which are invisible, and the residue P which we do per- 



24 INDUCTIONS FROM PERCEPTION. [Book II. 

ceive is bound up with other circumstances II, the results of 
conditions which have no influence on the matter in ques- 
tion though they are operative at the same time. A com- 
plete expression of the actual fact demands addition and 
subtraction and would run thus S + s — cr is P + p — 77 or 
2 is n, while our first defective observation set down S isP 
as the fact. Only for the complete proposition 2 is II (sup- 
posing this were given in a peculiarly fortunate perception) 
would universal validity be guaranteed by the law of iden- 
tity, — not for the incomplete proposition £ is P, which puts 
together what is not really thus connected. 

253. It is important to bear this in mind, for apart from 
it we cannot understand a right, which science claims and 
which is essential to her development. If the question is 
as to a predicate IT, which we do not yet know and which 
we expect to find in a subject 2, then wherever we are sure 
that we perceive this subject 2 whole and complete, and 
nothing else but 2, we are equally sure that a single obser- 
vation, which acquaints us with II, has an universal import 
and that in every possible case, where the same 2 is re- 
peated, the same II must unalterably present itself. When 
the chemist is instituting an experiment, if he only can be 
sure that he is dealing with one definite substance and 
applying to it one definite reagent and has excluded the 
possibility of any foreign conditions influencing the result 
he is going to observe, then he never doubts that the re- 
action observed in this single experiment will exhibit itself 
identically whenever the same circumstances are repeated. 
He at once assigns to a single perception the rank of an 
universal truth. When the physicist undertakes a measure- 
ment he takes care first of all to eliminate the sources of 
error, with which, as he well knows, he is beset, but when 
once he has purified his observation he never dreams of 
regarding the fact that it comes ultimately from sensible 
experience as a reason for accounting it valid merely for 
this one occasion. It never enters into his head that under 



Chap. VII.] 



GROUPING IN PERCEPTION. 



25 



similar conditions the same object might perhaps on another 
occasion have a different magnitude. We need not enlarge 
on this head. Once suppose that a single observed case is 
valid only for itself and not for its repetitions in like cases 
— that the record of an instrument is correct for the one 
occasion in which it is noted and not equally correct for a 
second occasion under identical circumstances — once sup- 
pose that with like object and like conditions a different 
result may be true, and there is an end to all possibility of 
developing universal truths from experience; there is an 
end not merely to the discovery of laws but to the use of 
the word 'law' with any intelligible meaning. The art of 
induction, which is to bring us to universal laws, rests 
wholly on the acumen shown in developing pure and self- 
connected propositions of the form 2 is Ü out of the im- 
pure and confused material of our perceptions, which come 
to us in the form S is P. 

254. Let us try to sketch in a connected series the steps 
essential to that development. The countless impressions 
which we receive in succession or together may be taken as 
an indiscriminate mass 12. How do we come to distinguish 
in the mass of them groups A, B and C and to regard each 
group as a self-connected perception? It is because the 
impressions we so gather into one are permanently con- 
joined and thus raise themselves above the level of the 
shifting background, or again because by their simultaneous 
appearance at intervals they contrast with the uniform 
background formed by the others. This first step is a 
necessary one, but there is so far no act of thought. 
The mechanical course of our ideas is the agency 
which singles out these perceptions and first makes them 
objects of our involuntary attention and of our future 
thinking. And the result thus reached is proved by subse- 
quent consideration to be wrong as often as right. The 
really connected is too often but incompletely conjoined 
and mixed with that which in no way belongs to it. We 



36 



INDUCTIONS FROM PERCEPTION. [Book II. 



are impelled in like manner to the second step which 
consists in splitting up the sum of impressions contained 
within each of the gronps A, B and C and in taking one 
part of each sum as a subject and the remainder as a 
predicate attaching itself to the subject. It is our psychical 
mechanism which accomplishes the beginning of this step 
also. Thought indeed actively intervenes before long and 
intensifies this mere conjunction of two ideas — this mere 
adherence of one idea to another — by transforming it into 
an objective connexion and by establishing an opposition 
between the subject and predicate, between the former as 
essentially independent and the latter as dependent and 
simply attributed. Still it is the mechanical course of our 
ideas which always guides us in the application of this 
added principle of thought and which settles in each case 
which group of impressions is to rank as subject and which 
as predicate. 

Thus articulated the whole content of a perception A or 
B might now be expressed in the form of a judgment, — 
but a singular judgment only. The subject which we here 
distinguish as s 1 or s 2 is nothing but a perfectly determinate 
group of single impressions, such a group being the sole 
possible object of immediate perception, and the thought 
that either s 1 or s 2 may be taken as the example of an 
universal conception *S is an added thought. It cannot 
originate in an individual perception, but only in the com- 
parison of many which soon begins. And here again, when 
out of the several subjects s 1 s 2 . . . the smaller group of 
attributes common to them all is forced upon our attention 
till it emerges as a general picture, which is now denoted 
by the name S and which takes the place of those several 
subjects in our memory, — this also is at first the result of the 
course of our ideas which is conditioned in a mechanical 
manner by universal laws. Here too thought adds a new 
depth, it transforms the general picture, which only repre- 
sented what was found common to the various individuals, 



Chap. VII.] INDIVIDUAL PERCEPTIONS, 



27 



into a general conception which has the force of a law and 
joins what is essentially connected : but still it is the course 
of our ideas that determines the first applications of this 
added thought, and settles for us which elements of the 
subjects we compare are to be united in the general picture 
or again in the conception and which are to remain ex- 
cluded. The elements which are felt to be modifications 
of one universal and are at the same time more lively as 
impressions are accepted, and those which excite our in- 
voluntary attention less strongly or reciprocally destroy one 
another are rejected. 

And the result of this process is on the whole more often 
right than wrong. This is not the place to pursue in 
greater detail the psychological development of our con- 
ceptions, but there is one point worth mentioning. In the 
sensible impressions, which are the ultimate components 
of every perception, it is from the first not the differences 
of the actual impressions that are of prominent importance 
so much as the differences in their relations and in their 
manner of union. The mere child can neglect the differ- 
ences of colour and sees that the characters shown him in 
red are the same as those he has learned in black. His 
general pictures of trees and animals are drawn as yet but 
from few examples, but they already comprehend the 
essential traits with such accuracy that when he afterwards 
perceives a new and unusual shape, it readily takes its place 
in the series. Errors however do occur : how these are 
corrected is what we have now to consider. We have tried 
in the foregoing to make clear our starting-point, which is 
this : — the inductive process deals with individual percep- 
tions and its object is to establish a further connexion 
between them, but these perceptions are more than mere 
impressions that we passively receive. On the contrary they 
have already been so far worked upon by thought that not 
only have their contents already been divided by us into 
subject and predicate, but besides that we have already 



28 



INDUCTIONS FROM PERCEPTION. [Book II. 



brought the subject under the head of a general conception 
aS or at least meet it with a selection of such conceptions 
under one of which we try to bring it. 

255. Let us take the last case first. Suppose that a 
singular subject s m , new and not yet known, is presented in 
a perception through the sensible impressions p m q m r m , and 
suppose we have a perfectly clear image both of these 
particular marks and also of their conjunction. The image 
if we do not go beyond it does not contain one doubtful 
element and yet it does not satisfy us. We cannot rest 
until we know what the new object may be called, whether 
animal or plant. Our desire to know this is based on a 
twofold interest. Pure thought is interested ; for it is only 
by subordination under a general conception that the mere 
coexistence of the observed characteristics is transformed 
for thought into a well-grounded coherence. But what 
weighs much more with us is the practical wish to go 
beyond the observed fact and to justify ourselves by the 
general conception in filling in what we have not observed. 
For the name plant or animal would be for s m a. barren title 
had we not reserved the right to found upon it a claim 
to a number of properties as belonging to s m , which no 
immediate perception has shown to belong to it. We con- 
stantly find ourselves in this position towards real objects. 
For every perception howsoever accurate, let it even appre- 
hend every single mark that s m now has, is limited as to 
time. No perception can tell us the future with the present ; 
it can never say what our object will eventually do or 
become, and can only seldom and incompletely show us 
how it will alter with altering conditions. The gap that 
perception leaves in our mind we fill up by subordinating 
the observed s m to the conception of plant or animal. 
These conceptions have sprung from countless observations 
and they comprehend the whole collection of coexistent 
attributes, which can scarcely ever, and the successive 
attributes which can never be exhausted in a single per- 



Chap. VII.] 



INCOMPLETE ANALOGY. 



29 



ception. But it is only by the union of all these that we 
can adequately determine the real nature of s m , while a 
single perception of it gives us only an inexplicable fragment 
of a connected whole. 

This process on its formal side is an incomplete analogy, 
and since so considered it argues ad sub alter na7item from 
the observed sameness of some attributes in £ and s m to the 
sameness of all, it must be pronounced invalid by the canons 
of pure logic. None the less our whole daily life depends 
on the application of this incomplete analogy. We rest 
secure on it in dealing with substances useful or deleterious; 
it alone persuades us of the existence of minds like our own 
and assures us that their actions flow from inner motives 
such as we feel. And in fact our dependence on it scarcely 
needs to be mentioned, so plain is it. The practical 
question is as to the means by which the bare probability 
of these inferences can be made to approximate in value to 
certainty. The sameness of all the marks is what logically 
justifies the subordination of s m to S, and the natural con- 
clusion from this fact is that the probability of a subordina- 
tion being correct rises as the number of identified marks is 
increased. But it is evident at once that the value of this 
conclusion is much lowered by the necessity of taking into 
account the difference in value which exists among these 
marks. And this is not the whole difficulty. It is idle, it is 
a mere form to bid us direct our attention to the likeness 
above everything else of the essential marks, when as yet we 
have no means of distinguishing them from the non-essential. 
It is experience and experience alone which enables us to 
distinguish, and the few general rules we can lay down are 
all drawn from experience. There are attributes, which 
arise indifferently from the operation of widely-diffused 
causes on objects vastly heterogeneous, and consequently 
these marks cannot serve as criteria because not character- 
istic of the distinctive content, which any one conception 
connects, and again the modifications of these marks can 



30 INDUCTIONS FROM PERCEPTION [Book II. 

produce no essential difference in any such content. But 
how should we know this except from experience ? How 
but from experience should we know that mere quantitative 
differences in the marks are in the main but of slight 
importance and that diversities in the forms of conjunction 
and in the respective positions of those points in which 
relations centre are a matter of far greater moment ? There 
are universal conditions in the world, which tend to produce 
similar alterations in different substrata which they en- 
counter, and this gives us a test for the real and exclusive 
peculiarity of any kind of things. For each genuinely 
peculiar kind by reason of the specific mode, in which 
diverse centres of relation are united in it, exhibits unex- 
pected forms of reaction against the universal conditions. 
Thus specific reaction against non-special conditions is the 
sign of a genuine kind — but this sign is the teaching of 
experience. And it is experience once more that informs 
us that these rules have their exceptions in the case of some 
object that we perceive. There may be marks that seem 
unessential and whose variations may appear of no moment, 
and yet there may be in such trifles the sign of a radical 
difference pervading the whole nature of two groups of 
attributes connected under two different conceptions. 

In conclusion we must not forget that in trying to range 
new objects of perception under old universal conceptions 
we are not unfrequently driven by utterances of the same 
experience to quite an opposite result. These universal 
conceptions themselves — animal, plant, body — are altered, 
their boundaries are widened or contracted as our knowledge 
of things grows. On the one hand we may find in doubtful 
instances, which seemed to fall under one of these concep- 
tions, points in which their habit permanently differs from 
that of the undoubted instances ; or again instances which 
seemed to be excluded may be found to exhibit a gradual 
and uninterrupted approach to what is the character as a 
whole of the known and undoubted species. Thus it will 



Chap. VII.] 



IMPERFECT INDUCTION. 



3 1 



be seeh that we trust not to universal logical rules but to a 
knowledge of things for the correct carrying out of the 
imperfect analogy, by which we class a perceived object 
s m under a general conception S. In fact we must dis- 
tinguish applied logic itself as a theory of science from its 
application as a scientific activity. The theory can do no 
more than lay down general points of view, of which we 
should never have become conscious had we not exercised 
the activity. Logic therefore cannot step forward and claim 
to impose its rules on the whole domain of real thought, as 
if the whole work of the latter was about to begin for the 
first time ; it is of no use to the mind, which has as yet no 
conceptions at all, but only to the mind, which is already in 
possession of a manifoldly articulated world of ideas, acquired 
through its own experience or by tradition. No doubt much 
interest attaches for the psychologist to the task of explain- 
ing how all these conceptions have arisen, but this task does 
not fall to the theory of science. Its role is only to establish 
what is true and certain in these ideas now we have got 
them ; and in as much as many errors and defects must still 
cling to these rough and ready results of long intellectual 
development, theory must also point out how these short- 
comings may in future be remedied and that which is still 
doubtful be brought near to certainty. 

256. Now if, as we at present assume, the individual per- 
ception is so far logically formed, that we at once apprehend 
the particular object s m which it portrays as an example of 
a general conception S, it will engage our attention no 
further provided that we find in s m none but marks which 
belong to S, no matter whether all or only some of them. We 
shall however be led on to take a fresh step, when we find 
bound up with s m in the perception a mark M, which does 
not belong to the group of attributes conjoined in S, so far 
as we know S. Experience (here, as I said just now 1 , the 
only authority) teaches us to discriminate three possible 
1 [Section 255.] 



32 INDUCTIONS FROM PERCEPTION, [Book II. 

cases. In the first place altering conditions or accidental 
circumstances may have temporarily invested the s m of our 
perception with a property, real or merely apparent, which 
under other circumstances it would not display. Wise with 
the knowledge we have already won we quietly neglect 
many points of this kind : the same object wears different 
aspects according to its position, attitude, movement, 
distance or illumination, but we do not allow such differ- 
ences to shake us in our conviction of its identity and its 
agreement with the general conception £ ; cases where there 
is more room for doubt we decide by trying to make 
observations of the same object under different conditions ; 
it is only an M which adheres to it under all circumstances 
that is regarded as a permanent mark of its nature. But 
we still leave it unsettled, whether this M owes its presence 
to the individual nature of this subject s m , which after all 
remains a particular subject, or to the universal nature of 
the general concept S, of which the observed s m is a species 
or an example. To decide between this second and third 
case we resort to what is called imperfect induction \ that is 
we compare s m with other examples s\ s 2 . . of the same uni- 
versal £ with which we are familiar. In most cases what 
leads us to make the induction is that a number of indi- 
vidual perceptions s*-M, s°M, s 3 M thrust themselves one 
after the other on our notice, so waking in us a suspicion 
that the ground of M is universally to be found in the 
nature of S, in various examples of which we observe it. 
This presumption is rebutted, if we find a single subject s q , 
which has not the mark M when placed under the same 
variety of conditions, under which that mark attached to 
the subjects of the other perceptions. On the other hand 
all instances of S which have so far been accessible to our 
perception may possess this predicate M, without our pre- 
sumption in favour of the truth of the universal proposition, 
all S are M, being ipso facto corroborated. For when we 
argue that what is valid in a number of particular cases 



Chap. VII.] ROOT OF ANALOGY AND INDUCTION. 33 



however large is therefore valid universally we always argue 
ad subalternantem and such inference is to the last unsound. 
Still placed as we are we must hazard such inferences, for 
even if perception could embrace all existing examples of a 
genus, those that are yet to be will always evade our senses. 
Here too then all we can do is to heighten so far as we can 
the probability of this imperfect induction. In order to this 
we shall find two rules of kindred import of great use. In 
the first place the individual subjects from the observation 
of which we start must be very numerous ; the greater the 
number of such s the more manifold must the outward con- 
ditions be which act upon them, and of which we thus 
eliminate the force and influence. Any M which all these 
subjects retain in common in spite of such variety of con- 
ditions must owe its presence to no extrinsic causes but to 
the intrinsic nature of these subjects. Secondly we must so 
choose the subjects we observe that their specific or indi- 
vidual differences shall be the greatest possible within the 
limits of the genus or the species, the universal *S ; we thus 
eliminate the influence which particular resemblances 
between the observed subjects, which are independent of the 
nature of the universal S, might have in producing the com- 
mon predicate. The M which they all unite in retaining in 
spite of these differences will have its ground solely in the 
character of the genus itself, and the universal proposition, 
all 5 is M, which we wished to arrive at, will thus be justified. 

257. Pure logic raised a distinction between analogy and 
induction. If two subjects agree in respect of several of 
their marks, we gather that they will agree in all. This is 
reasoning by analogy. We make an induction on the other 
hand, when we argue that because several instances of a 
kind behave in a certain way, therefore all instances of the 
kind will behave so. We have used the words in the same 
meanings here, and it was accordingly an induction, by 
which we drew from the given premises s^M 9 s 2 M. . . the 
universal conclusion SM. But this procedure may be re- 

Logic, Vol. II. D 



34 INDUCTIONS FROM PERCEPTION. [Book II. 

garded more simply. Suppose we have made a number of 
observations and have found that all their individual subjects 
s 1 , s 2 . . . agree in possessing on the one hand all the marks 
belonging to an S, on the other hand the one mark M\ 
we may then conclude immediately by analogy that every 
subject s q , even though we have not observed it, will yet, 
provided only it like them possesses all the marks of an S, 
possess also the particular mark M. By such an analogy is 
it that we supply the premises s m M y s n M . . . not given in the 
perception, the subjects of which premises together with the 
subjects of the s^M y s 2 M . . . , which we have observed, ex- 
haust the whole compass of S. The business of induction 
then consists in merely summing up the single propositions 
thus either given or supplied in the universal proposition : 
all £ are M. We see from this that it is hardly worth while 
to separate in such applications of logic the part played 
by induction from the part played by analogy ; nor 
is it worth while to find fault with the loose usage which 
confounds the two expressions ; it is useless in short 
to try to refer to simple types of pure logic all the opera- 
tions of thought, which may be broadly included under 
the name of an inference. One who has time to waste may 
perhaps enquire whether a voyager, who has sailed all round 
a land, concludes by induction, analogy, or subsumption, 
that it is an island. What does interest us here is rather the 
question, how we arrive at any universal proposition Tabout 
triangles. We prove Z'by constructing the triangle s 1 ; but 
this triangle, which we thus set before our eyes, is always a 
particular triangle, never more. It would seem as if T can 
in the first instance be true only of it, and always true of it 
so often as we construct it in the same way. Now we may 
of course change our mode of drawing it ; still even if we 
found the proposition ^corroborated in a thousand different 
triangles s\ s 2 , s z , . . , this number would dwindle to nothing, 
when compared with the infinite number of possible 
triangles, which we have no opportunity of testing. 



Chap. VII.] ONE THE SYMBOL OF ALL, 



35 



It is not therefore by any summing up of particular per- 
ceptions, which we create for ourselves in these construc- 
tions, that we reach the conclusion that 7"is true universally 
of all triangles whatever. We must be justified in regarding 
each single triangle we draw as a symbol for all, so that what 
holds of it holds of all the others which it represents. And 
our justification does not lie in the peculiar nature of spatial 
perception ; that merely supplies the content of T\ it does 
lie in the fact, that we only pay attention in our reasonings 
to those features, characteristic of the triangle drawn, which 
we have produced through the very process of constructing 
it, that is, to its property of being a plane figure, included 
by three straight lines. The figure actually drawn can never 
exhibit this property in the abstract and apart from other 
properties. It can only do so by having sides of definite 
length and a sum of angles distributed in a certain way. 
But we do not let these special qualities have any influence 
on our conclusions ; suppose we have unintentionally con- 
structed our triangle with right angles or equal sides, we 
shall set aside propositions which are valid because of these 
qualities and these alone, as having nothing to do with the 
subject of predication which we had in mind. Psycho- 
logically no doubt it is the unanalysed impression of re- 
semblance, which prompts us at once to transfer to all 
triangles by analogy the proposition T proved of the 
particular triangle we have drawn. Our logical justification 
for doing so is twofold : first it lies in our consciousness 
that all triangles, whether already constructed or no, may 
still be classed under the universal conception of a triangle, 
which conception we have symbolised in our construction ; 
secondly in the consciousness that in that single symbolic 
representation we saw the proposition in question flow solely 
from this conception without any appeal being made to any 
other conditions. 

258. In attaining to universal propositions of the form 
all S are M ) induction has reached its first goal, and it is 

D 2 



36 INDUCTIONS FROM PERCEPTION. [Book II. 

possible to rest content with the result, especially when we 
are dealing with a question of practical life; for in such 
questions w 7 e can go without a reason, so long as we are 
certain that as a matter of fact M is really true of all 
instances of S, say of all men; we do not care so much 
to know why it holds of them, and why only of them and 
not perhaps of animals as well. The theoretic impulse 
however is not satisfied with merely joining M to its 
proximate subject ; it would fain seek out within the limits 
of £ the narrower group of attributes, which contains the 
ground of this conjunction, and which conditions the same 
attribute, wherever it may occur, perhaps even outside 5. 
Then the induction is pushed further ; we use a series of 
universal propositions of the form : SM, RM, TM ... as 
our new premises and try to deduce from them an universal 
conclusion of the form all 2 are M. In this new r conclusion 
we understand and denote by 2 the true subject or the 
conception of the genus, or, to put it in another way, that 
complex of attributes, on which the predicate M in all cases 
depends and from which it results. Thus in our first in- 
duction we shall reach the proposition SM: in all mammals 
an exchange of gas takes place in respiration ; in a second 
induction, in which S is successively replaced by birds, 
fishes, and amphibia, we shall reach the conclusion all 
animals require an exchange of gases. This new conclusion 
at once throws light on the earlier one, by showing that 
what we had hitherto only observed as an isolated fact is 
really necessitated by the universal nature of animal life ; 
a third induction sets alongside of ^M a new premiss to the 
same effect, viz. all plants display though in another way 
the phenomenon of a change of gas ; its conclusion 2W, 
all organic beings whatever find themselves in like case, 
shows us the phenomenon in question bound up with a 
still more universal subject; and lastly by comparing the 
behaviour of bodies which resemble organic bodies in 
structure towards the surrounding atmosphere we might 



Chap. VII.] HYPOTHETICAL JUDGMENTS. 37 

be led to the thought that under the conditions prevalent 
on the earth's surface, such an exchange of material is 
absolutely necessary to the development of those interde- 
pendent processes of change, which make up organic life. 
In all this it is to be noticed that the further we advance 
these inductions, the less do we care to obtain as our result 
a categorical judgment of the form S is P ; we are no longer 
seeking the highest general conception, to which a given 
phenomenon attaches a predicate ; what we are in search 
of is a hypothetical judgment, which will acquaint us with 
the most general condition C, upon which the phenomenon 
always depends and of which it is the consequence when- 
ever it occurs. And this new problem of discovering the 
conditions under which the content of a perception coheres 
is of such preponderating importance in applications of the 
inductive method, that we shall henceforth in our investi- 
gation of that method confine ourselves to the form which 
it assumes in order to the solution of that problem. 

259. Let C and E respectively denote two groups of 
observed events ; we will suppose that something or other 
in the way in which they appear has awakened in us the 
presumption — to be subsequently confirmed or corrected — 
that the two groups are really connected, and that C is or 
contains the cause of E, and that E is or contains the effect 
of C ; lastly, let us bear in mind the remark we made at the 
beginning of this chapter (sect. 252), to the effect that C will 
seldom really contain the full cause of E and nothing but it, 
E seldom contain the whole effect of C and nothing but it. 
We may then indicate our problem thus : to discover from 
the impure observations C E the pure case B E L of two 
terms belonging together of their own nature and related 
as condition and conditioned ; and if we have to define the 
conception of this pure case we shall say, that in it B is 
the adequate ground of 7? and the ground of nothing else 
beside E y while on the other hand F is the full consequence 
1 [' Bedingung " and £ Folge.'] 



38 



INDUCTIONS FROM PERCEPTION. [Book II, 



of B without being at the same time the consequence of 
any other ground. In applying this definition we may 
abate somewhat of its rigour according to the different 
interests which from time to time rule our investigations. 
For instance we may be content to know that B as often as 
it is given produces F, no matter whether it conditions 
anything else besides F, or whether F can be produced by 
any other antecedent as well as by B. We shall only be 
content with such a result, however, where we are merely 
concerned to know the real causes which produce the effect 
in question, When the question is not as to the real causes, 
but a theoretical question as to the ground 1 , owing to which 
these causes condition that effect as their necessary conse- 
quence, we shall always have to determine B and F with all 
the precision aimed at in the definition ; even where F may 
be due to different but equivalent causes it is not that in which 
these differ that is the true ground of this consequence, 
for each cause has besides F other and separate effects ; 
only what is common to all of them is the true ground 
B, and this B has then no other consequence than just 
this F 

260. From a single impure case C E the pure case B F 
can only be guessed by an accidental and happy inspiration : 
it can be demonstrated with certainty only by a comparison 
of several different cases. If we can observe a sufficient 
variety of cases we shall be able to detect elements, which 
do not really belong to each other and are merely accessory, 
by the variety and change of the relations they bear to one 
another. We can then let these unessential elements drop 
away and retain in its pure form the pure relation B F, 
which they all involve. These impure cases form the raw 
material, on which we go to work, and are mainly supplied 
by observation ; but the course of things if left to itself 
presents us in but few fields of research with the full 

1 [Cp. ' Metaphysic,' sect. 51, on the distinction between 'Cause' 
and ' Ground ' or ' Reason.'] 



Chap. VII.] EXPERIMENT AND OBSERVATION. 



39 



number of cases that we should require in order to com- 
plete that elimination. It requires long epochs far trans- 
cending a single individual's range of observation for many 
natural processes to unfold the whole series of aspects which 
one should know in order to grasp the coherence of their 
conditions. But there are other obstacles to observation 
besides length of time ; in the case of many actual products 
it is impossible to see how they have become what they are, 
because they never of themselves permit of being regressively 
analysed into the conditions out of which they arose. It is 
not often that we find ourselves so favorably situated as in 
the case of astronomy. This science has never met with more 
than accidental obstacles in its accumulation of numberless 
data in regard to an interdependent and periodic play of 
events. Yet even astronomy requires, in order to satisfy all 
its wishes, to be supplemented by observations of the past, 
and of these it finds but an inadequate supply. 

Wherever we can by our own agency influence the object 
we are investigating we can remedy this want by experiment. 
We can institute at will a certain group of conditions C, and 
so compel the causes which are really at work to respond 
with an effect E, which would otherwise perhaps have 
never come within the domain of our senses. By varying 
at will the quantity and composition of that C we can 
bring about in E a series of changes in quantity and kind, 
which were still less likely to offer themselves unsolicited 
to our observation. Again we can break up C into its 
component parts, and in each experiment allow but one of 
them, or a definitely assignable group of several of them, to 
take effect, at the same time cutting off the rest from action. 
The constituent elements of the result E admit of being 
separated in the same way, so that we learn which of them 
depends on which element of the compound C. Thus 
experiment is the practical means by which we furnish 
ourselves with observations in such number and involving 
such mutual differences and affinities, as is requisite in 



40 



INDUCTIONS FROM PERCEPTION. [Book II. 



order to the elimination of what is unessential in them and 
the derivation from them of a pure case B F. Denned in 
this way it is clear that experiment only has an advantage 
over observation in so far as it is capable of supplementing 
the usual deficiencies of the latter ; its function is to furnish 
us with suitable and fruitful observations instead of the 
unsuitable and unfruitful ones which offer themselves. But 
it would be perverse to ascribe to it any other and mystical 
use in addition to that ; we cannot set it over against obser- 
vation as a new method of knowledge ; it is merely a way 
of preparing and setting before ourselves phenomena which 
it is of importance that we should observe. And for the 
same reason experiment must not be unqualifiedly set above 
mere observation. In our day it is a prejudice of half- 
culture to suppose that anything that can be observed in 
broad daylight, without any preparation, ready to hand and 
in the gross, remains as a matter of course open to question ; 
that alone is supposed to be true which can only be per- 
ceived in microscopic fashion, on a minute scale, after all 
manner of preparation and under conditions which render 
it very difficult to perceive at all. Such an assumption is 
paradoxical, and if elevated into a general principle becomes 
absurd. Still it is a just assumption to make in certain 
cases. In particular we can only ensure accuracy in our 
determinations of quantities by such artificial means, never 
by coarse observation alone. We must grant all this, but 
conversely observation often acquaints us with broad 
characteristics of phenomena, which in experiment would 
have been obscured by special conditions. 

261. I shall now attempt to lay before my reader the 
various kinds of relation between C and E, with which 
observation and experiment acquaint us, not in the hope 
of exhausting them, but in order to illustrate by examples 
how many and various are the possible cases and the con- 
sequences to be drawn from them. 

i. The case (+ C E). C and E may be continually 



Chap. VII.] COEXISTENCE AND CONCURRENCE. 41 

present in reality and continually together ; still their mere 
coexistence, however uniform, does not warrant our con- 
cluding that they are so related as to condition each other, 
though such a relation may perhaps exist between them. 
Iron and silver and all the other chemical elements are 
always present in the world together ; but it does not follow 
from this that one of them is the condition of another's 
existing, or that all of them collectively are joint effects of 
a single cause. At best the philosopher, for certain specu- 
lative reasons, which we cannot enter into now, may 
question the possibility of there being a plurality of 
elements coexisting yet unconditioned in any way by one 
another. But the primary use of induction is to understand 
nature, and the scientific understanding refuses to accept 
mere coexistence as evidence of an ulterior connexion. We 
find, moreover, that in each single one of these elements 
various properties or modes of action are uniformly com- 
bined. For example, all have in common the property of 
gravitation, and each in particular has as well its special 
affinities to all the rest. This case is not the same as the 
last ; here we have one and the same subject, with different 
properties coexisting in it as its own. This oneness of the 
thing forbids us to suppose that the several attributes found 
in it have no reference to one another, and there is thus 
awakened in us the natural tendency of thought to explain 
one of these properties by the other or both by a third 
original one, which under different conditions transforms 
itself into those two. We will not at present fix the goal 
to which this logical impulse may lead us in the future; 
for the present it remains just an impulse which leads to 
nothing so far; for so long as our observations reveal to 
us nothing else than the perpetual coexistence of both 
predicates, they do not supply us with the means of 
showing the nexus of conditions, which perhaps exists 
between the two or between them and a third. 

2. The case (+ C+£). C and E are present together, 



42 



INDUCTIONS FROM PERCEPTION. [Book II. 



not uninterruptedly, but in frequent recurrence. In such 
a case it may be mere coincidence that brings them to- 
gether without there really being any reciprocal connexion 
between them, each resulting from a separate condition. 
This is what we shall conclude with regard to the many 
mischances which befall us on Friday, and with regard to 
countless other superstitions of the sort. But we do not 
acquiesce in such a conclusion, if we can conceive of any 
real connexion between the C and E thus found together, 
and if there is any hope of finding out their connexion. 
We never think of acquiescing in it unless we shortly after- 
wards learn from further observations that their association 
is quite exceptional and abnormal. In itself the hypothesis 
of mere coincidence is the least probable of any ; whenever 
phenomena occur frequently and repeatedly together there 
is probably a causal relation; it only remains difficult to 
decide whether one of the two C and E is cause or effect 
of the other, or whether both are not mere co-effects of a 
third cause Z. This doubt remains even when C and E 
appear not simultaneously, but after one another in a 
definite sequence in all cases of repetition. In that case 
C may no doubt be the cause of E, but both may also 
be joint effects of a third Z y which is not uniform, but 
undergoes changes, which succeed one another in a 
definite order. Day and night always follow on each other 
in this way, yet they do not produce one another, but are 
successive joint effects of the earth's revolution on its axis. 
Lastly, it may happen that E has always remained un- 
noticed, and only meets our observation when C occurs; 
thus the heart always beats in a living person, but a healthy 
man hardly ever feels it, unless a special excitement C 
supervenes ; then C is not indeed the cause of E, but 
the condition of its meeting our observation. 

3. The case (— C+E). Doubts are left unsolved by the 
last case, which can only be settled by further observations 
which present themselves or are procured by experiment. 



Chap. VII.] 



CESS ANTE CAUSA — . 



43 



We may find that E also occurs without C, or that C may 
be experimentally suspended, without at the same time sus- 
pending E. In such a case we cannot of course conclude 
that C is not the cause of E, though we may conclude that 
it is not its sustaining cause. The former conclusion would 
be a hasty one ; we should appeal in justification of it to 
the principle : Cessante causa cessat effectus, but the only 
clear meaning which can be given to this principle is that 
with the cessation of a cause will cease those effects which 
the cause would have continued to produce had it continued 
to exist. That effects once generated are not all alike in 
this respect is shown by the simplest examples ; a move- 
ment continues after the shock has ended, which produced 
it ; the boiling of water ceases when the supply of heat 
abates, which produced it and is required as its constant 
sustaining cause. A child does not die with the death of 
its parents, the sole causes of which its existence could be 
the effect; but the equilibrium of a supported weight is lost 
when the supports are withdrawn. We need not at present 
analyse these cases any further ; we can refer them all to 
an universal law of persistency 1 , which in reference to 
our present problem we thus express : every reality, which 
has once been produced, of whatever kind it may be, con- 
tinues to last, unless counteracting causes annul it. The 
effects which a cause produces do not therefore stand in 
need of a cause to sustain them so long as they consist 
in states of a subject which are in equipoise not only with 
the permanent nature of that subject, but with the external 
conditions under which it exists. They do require such a 
sustaining cause however, if there prove to be either in 
that nature, or in those conditions, forces which by their 
influence tend to transform it. If therefore E lasts on 
after C has ceased to be, three cases are possible : either 
there is no causal interdependence between the two at all, 
or else C is indeed the cause which produced though not 
1 [Cp. ' Metaphysic/ sect. 162.] 



44 



INDUCTIONS FROM PERCEPTION [Book II. 



the cause which sustains the effect E, and in this case again 
C is either a productive cause alongside of other productive 
causes, or it may be the sole cause capable of producing C. 

4. The case (+C— E). C may be observed to occur 
without being followed by E, but there is of course nothing 
in that to attract our notice, unless it conflicts with our 
usual experience, i. e. unless C and E have been observed, 
as a rule, to occur in conjunction with each other. In 
such a case it may be that C is not the cause of E, and 
we then, by drawing this new conclusion, correct the earlier 
one, which we had formed from our observations to the 
contrary. The connexion of causes and effects in reality 
however is not the same as that between ground and con- 
sequent in the field of abstract thought. There every 
ground, which holds at all, bring about its consequence 
not partially but wholly, and also in such a way that the 
whole of it can be perceived in the result. Two grounds 
may be operative at once, e.g. a quantity g may have as 
its conditions two equations determining its relation to x and 
must satisfy both. The influence of the second equation 
will then always show itself in this way, that of the many 
values of g, which the first alone left possible, it will leave 
over but a single one or a definite number of these con- 
joined in a regular manner. A change E on the other 
hand, which must follow in reality from the cause C, can 
always be set aside by a countercause Z so that it is lost 
to perception. We cannot say that Z annihilates the 
capacity which C has of producing an effect, for C can 
only be restrained from producing its effect E, so far as it 
reacts itself on the restraining Z; in this Zit always brings 
about another effect E\ instead of E, which we expected, 
or it assumes itself, under the joint influence of Z and of 
its own tendency to produce an effect, a state E l , which it 
would not otherwise assume. But this E 1 is very often of 
such a nature as wholly to withdraw itself from direct ob- 
servation; in that case E seems to be altogether absent, 



Chap. VII.] 



NEGATIVE INSTANCE. 



45 



while C is present ; as a matter of fact E has only changed 
its form. This is invariably the case when moving forces 
meet with a fixed obstacle; they then seem to have no 
effect, whereas they really exert a strong pressure on the 
resisting body. If then we find that E fails to follow on C y 
it may be of course that there is a want of any causal con- 
nexion between the two at all, and in that case we must 
put a different interpretation on the sequences of C and E, 
which we have observed. But C may also be one or even 
the sole cause producing E and yet be prevented from 
bringing about E by a counterforce Z. This shows how 
much need there is of being circumspect, of looking round 
in every case of the sort to see, whether in place of E we 
cannot discover an effect E l , which but for the obstacle 
it reveals would be absent. Lastly, when we institute C 
experimentally, and do not find that E ensues, and at the 
same time can find no trace of a E 1 taking its place, we are 
justified in concluding that C is not a cause capable of 
producing E at all. 

5. The case (-C—E). So far as mere observation goes 
the simultaneous absence of C and E will seldom strike us 
as noticeable, and when it does it will be because it con- 
flicts with what we remember experiencing in the past. If 
however C has been constantly present and we find that 
when it ceases to appear E vanishes as well, the most 
natural thing to suppose is of course that C is at least the 
condition which sustains if not the condition which pro- 
duces E, or may be that C and E are both joint effects 
of a third cause Z, and they both vanished because it has 
ceased. If E ceases to appear when we suspend C ex- 
perimentally, the former alone of the two alternatives seems 
possible ; still it may be otherwise. When we talk of a 
cause which has been active hitherto being suspended we 
may mean something more than that it just ceases. To 
effect such suspension we often have to take positive pre- 
cautionary measures, and the new influence thus brought 



46 



INDUCTIONS FROM PERCEPTION [Book II. 



into operation may, while suspending C, at the same time 
create new conditions paralysing the further action of 
causes, to which though quite distinct perhaps from C, 
the presence of E was all along really due. Such new 
conditions would result in the suspension of E as well. 
There was a prolonged controversy between those who 
maintained that infusoria are generated from an infusion 
of organic matter without germs of their own kind being 
present beforehand, and those who contended that their 
generation was conditional on the presence of spores or 
seeds adhering to the organic substance itself, or conveyed 
by the atmosphere, or contained in the water. The only 
way to settle the dispute was to show that the generation 
E of infusoria ceases when all access C of spores or seeds 
capable of producing life is cut off. They cut it off by 
boiling the water along with the organic substances and 
introducing air through red-hot tubes. The use of such 
means no doubt ensured the absence of living germs from 
all the three bodies concerned in the result ; at the same 
time they were so violent that in excluding the germs they 
might also have rendered inoperative the causes which the 
counter theory assumed, viz. the inherent capacity of organic 
matter of developing into living organisms. The experi- 
ment therefore required to be modified in such a way as 
to eliminate the doubt. 

6. The case ( + E + C). In none of our conclusions 
thus far have we established more than that C is a cause 
of E ; that it is the sole cause, so that the converse of the 
proposition is true, and every E is the effect of a C, could 
only be ascertained by some method of exclusion, by which 
we could make sure that no other conceivable causes have 
the effect E. This exclusion is never possible with regard 
to the countless proximate causes, which are to be found 
at work in nature. We could not think of it till the 
elaboration of our perceptions was so much advanced as 
to have revealed to us a number of universally operative 



Chap. VII.] 



SAME EFFECT, SAME CAUSE} 



47 



forces, which could be exhausted in a complete disjunction, 
to some modification of which forces every result whatever 
would be wholly due. (Nevertheless inductive science fre- * 
quently arrives at such convertible propositions ; when in 
several cases it has found C to be the cause of E, it as- 
sumes that an JE, of which it does not observe the cause, 
is to be referred as an effect to the same C. Logicians 



cannot be gainsaid when they declare it to be wrong to do 
so according to the canons of formal logic. For it is quite 
clear that the particular judgment, many E are effects of C, 
in no way warrants our concluding ad subalternantem, that 
all E are effects of C. Nor can the hypothetical judgment, 
if C exists, E exists, be converted simply into the judgment, 
if E exists, C exists. But we would remind those who would 
lay too much stress on this fact that the scientific enquirer 
in drawing the conclusions here impugned does not pre- 
tend to be following the abstract ideal of a perverse logic. 
His knowledge of things and of the universal ways in which 
natural events do as a matter of fact usually occur, is so 
ample that he feels himself justified in making good any 
short-comings, which such conversion may have in respect of 
formal logical validity.^ There might be in nature, he would 
say, a hundred similar effects due to a hundred different 
proximate causes, only it is not so in fact ; as a matter 
of fact similar effects flow from causes which do not re- 
semble each other merely in being able to produce these 
similar results, but this ability itself depends upon an 
ulterior similarity between the causes. 

We hardly need dwell on this any longer. In order to 
make up for what our conclusions lack in point of mere 
logical cogency, we appeal to the fulness of the knowledge 
we have already actually acquired, and such an appeal must 
obviously carry with it enormous weight. On the other 
hand we must bear in mind that the justification so derived 
has its limits. Newton has expressed the principle in ques- 
tion in his second rule as follows : ' effectuum naturalium 



43 



INDUCTIONS FROM PERCEPTION. 



[Book II. 



ejusdem generis eaedem sunt causae.' I think we may 
without lessening our respect for his immortal genius confess 
that this formula by no means fulfils from a logical point of 
view those requirements of precision, which as a mathe- 
matician he knew so well how to satisfy. We do not forget 
that this rule is not put forward as a logical law, but just as 
a rule or practical maxim of natural philosophy, probably 
called forth by the brilliant discoveries which it preludes. 
But even as such it is not a little indefinite and every single 
one of its terms needs to be explained. To begin with, the 
words idem genus require to be defined, so that we may 
know what effedus naturales belong to the same and what 
to a different genus. I do not lay much stress upon that, 
nor can we even in logic altogether dispense with some 
such vague impressions ; for the rest we interpret it in this 
connexion in the sense that merely quantitative differences 
would not make a difference of kind between processes 
resembling each other in form. But what are effedus 
naturales ? If by these words we understand every natural 
event so far as it is referred as an effect to any cause what- 
ever, the whole proposition which ends with the words 
ecedem causcz is evidently untenable, so long as this last 
conception is left indefinite. If in the idem genus we 
include as we did just now quantitatively different results, 
these can only have causas ejusdem generis, not causas easdem \ 
the causes no less than the effects must differ from each 
other in respect of quantity. But this is not all ; the 
necessity of their being ejusdem generis is rebutted by the 
most common experiences, which teach us that causes may 
often differ widely from each other and yet be equivalent 
and bring about one and the same kind of effect. Suppose 
the velocity with which a body B approaches a point C to 
be uniformly accelerated, this much is of course clear and 
necessary, that some force must act on it able to produce 
this and just this effect ; but of how many different kinds 
may the forces be which do this ! They can act as a pull 



Chap. VIT.] CRITICISM OF NEWTON'S CANON. 49 



a fronte from the point C, they may also act on B as a push 
a tergo, so as to drive it towards C. The former mode of 
effect may be due to the forces of electricity or gravity, the 
latter to a series of self-accumulating shocks. If we persist 
in regarding all these causes as easdem or as ejusdem generis, 
because in spite of their essential differences in other 
respects they all agree in producing just this one effect, we 
not only use words in a very improper way, but we turn the 
rule into a trivial tautology. For it is obvious that all 
causes, which are to have effects of the same kind, must at 
least be so far themselves of the same kind as to be each 
and all of them capable of producing these effects ; they 
must therefore be equivalent as regards this effect. This is 
a_jmere deduction from the law of identity and as such 
requires no special maxim of physical science to enforce it ; 
such a maxim should it is evident represent something as 
in point of fact true, which on formal logical grounds is not 
necessary ; that is to say, in this particular case, such a 
maxim will assert that the causes of similar natural processes 
are not only similar in reference to these events, but are also 
similar independently of them. But as we have just seen, 
there are experiences which prove that what this maxim 
asserts cannot be universally true. 

There is still another sense however in which the words 
effectus naturales may be used ; they may mean not so much 
natural processes as processes in nature, that is to say not 
such events as incidentally arise on a petty scale out of the 
application of physical laws to fortuitous groupings of con- 
ditions, but such processes as have their abiding place in 
the grand theatre of nature, processes which would be con- 
sidered ends of nature by anyone, who felt himself justified 
in using this conception. There is nothing in Newton's 
language to force us to interpret it in this way, but that 
something of the sort was before his mind is probable 
from the prominent position he gives his rule in the intro- 
duction to a work, which was intended to embrace in one 

Logic, Vol. II. E 



50 INDUCTIONS FROM PERCEPTION [Book IL 

vast intuition those very abiding, all-embracing, and all- 
determining events of nature as a whole, which we have 
described, — we mean the revolution of the planets, the path 
our earth pursues, and the unceasing tendency of bodies to 
fall or press toward each other. Viewed in this light the 
above proposition would not be a direct rule to guide us in 
our investigations, but the expression of an actual fact, 
of which the existence has indirectly a controlling influence 
on the path investigation will strike out ; we mean of the 
fact that there are at work in the world not an infinite 
number but a very small number of highest and most 
universal mutually independent causes, to one of which 
every group of interrelated effects is in the gross to be 
referred, though in detail one and the same effect is not 
always due to the same cause but may be due to very 
diverse equivalent proximate causes. It would still be 
difficult to fix the line of demarcation separating those great 
causes from these petty ones ; nor would it be less difficult 
to make out what part of the proposition thus interpreted 
most deserves to be insisted on, that which points to the 
sameness in kind of the highest or that which points to the 
difference in kind of the proximate causes. Anyhow the 
scientific praxis of Newton is so admirable, that we do 
better to try to emulate it than to make a superfluous 
parade of its general maxims. 

I will return to an instance. A chemist observes that a 
particular element C yields the reaction E ; he then finds 
that a strange body, which he is examining for the first 
time, exhibits the same reaction E ; he infers from this that 
the body in question is C, and this inference so far from 
being based on the simple conversion of that observation 
rests on the consciousness, which he has, of having already 
tested all the elements to be found on earth, and of having 
got this particular reaction E from none of them except 
from C. This proof by exclusion is not in a formal 
sense absolutely safe, but yet carries with it great probability. 



Chap. VII.] CONVERSION AND EXHAUSTION 51 

If a new element C 1 is discovered, which gives the same 
reaction as C, he is so much the wiser, and forthwith looks 
about for some other test, by which to distinguish the two. 
Not quite the same amount of probability attaches to the 
conclusion drawn from spectrum-analysis. It is argued that 
materials, which in the spectra of the heavenly bodies 
produce the lines E, are identical with those terrestrial 
elements, which in a gaseous state display the same lines E 
in their spectra. Now we have not experimented with 
those non-terrestrial substances, and so we cannot be sure 
as we were sure in the former case that there are not 
several elements, differing in other respects yet agreeing in 
having this one reaction E. It is very probable there are 
not, because we know of no instance of one terrestrial 
element having the same lines in its spectrum as another 
without being the same element, besides which the bodies 
of our solar system may be regarded as connected fragments 
of what was once a single mass. Many bodies that are 
chemically quite different display the same colors in a light 
which falls upon or passes through them, and this proves 
that the capacity in question, i.e. the capacity of reflecting, 
absorbing, and transmitting different waves of light, does 
not cohere quite simply with the chemical nature of sub- 
stances. On the other hand two elements are not attended 
with peculiar effects E or E 1 , merely because the one is 
called or is Potassium, the other Sodium. The truth is 
that the only reason for their being or being called the one 
or the other is that the universal forces, with which bodies 
assert themselves against one another, occur in the two 
bodies in question with specific coefficients of different 
magnitude. But, it may be objected, there are conditions, 
which we cannot reproduce in any experiment. Under 
such conditions, — e.g. in the temperature prevailing on the 
sun's surface, — might not one of these coefficients, by the 
combination'of which one element is characterised, assume 
a value which under terrestrial conditions it would only 

e 2 



5 2 



INDUCTIONS FROM PERCEPTION [Book II. 



exhibit for another element ? The result would be that 
different elements might occasionally exhibit the same lines 
in their spectra. All this is not so utterly unlikely, and so 
we cannot banish all doubt from our minds as to the glimpse 
into the constitution of the heavenly bodies, which spectral 
analysis has vouchsafed to us. 

7. The case (-E — C) would agreeably to our use of 
symbols mean, that we argue backwards from the absence 
of an effect E, which in other observations we found to 
follow on C, to the absence of C. There is no need of 
further explanation ; all we can correctly infer from the 
absence of E is this, that although there may possibly be 
many different causes C\ C 2 , C 3 . . . , all capable of pro- 
ducing E, no one of them has been actually operative, 
either because no one of them existed or because each and 
all of them met with obstacles, which rendered the pro- 
duction of E impossible ; the latter alternative is settled as 
before, according as traces are or are not to be found of 
another effect E 1 , which takes its place. 

262. Now supposing that in one of the ways described it 
has been set beyond a doubt that C either is or contains 
the cause of E, this last question can only be answered by 
repeated observations and experiments, by which we shall 
test one after another the several elements of C and see 
what is the effect of each. We may have no difficulty in 
distinguishing these elements, or we may only be able to 
separate them by means of artificial arrangements. In order 
to this we substitute for the cause C and effect E two 
equivalent composite groups consisting of the events 
a + b + d and a + /3 + 6 respectively. The relations which 
result are manifold. The following are some of the simplest 
cases and will serve as examples. 

1. The case (C—a — E). The material analogy con- 
veyed by these symbols is plain. They signify that the 
absence or experimental suspension of one part a of the 
cause C produces no change in the effect E. If this be 



Chap. VILj 



TYPES OF INDUCTION. 



53 



really the case, if, that is to say, the E now observed be 
exactly the same as the E formerly observed, we shall 
naturally conclude and shall be perfectly justified in con- 
cluding that a has nothing to do with producing the effect. 
But this is just what we do not always find ; we are now 
dealing with all these cases simply with regard to the 
manner in which they appear to our observation, and we 
must remind the reader that very often the effect, so far as 
we can observe it, remains unchanged, whereas in fact the 
real effect has through the suspension of a undergone a 
change into E l . Suppose six cords of equal length are 
fastened to the corners of a regular hexagon, on which is 
slung a weight. If we then remove the first, third, and fifth 
cords, the weight will, if the remaining cords are strong 
enough, not only remain hanging, but wijl appear to main- 
tain its absolute place in space. Yet the latter is certainly 
not the case ; the tension of the three remaining cords is 
increased, and as they have stretched a little the weight 
itself has sunk slightly in a vertical direction, and herein 
consists just the new effect E\ which has taken the place 
of E \ the difference between the two is lost to a super- 
ficial observer ; who is led to conclude wrongly that the 
three other cords contributed nothing to the original effect 
E, whereas in point of fact the work which they did before 
has but been vicariously undertaken by the other constitu- 
ent parts of C. It is hardly requisite to notice how common 
an error it is to suppose, because an effect is so minute as 
almost to escape our notice, that it therefore does not exist 
at all. Such an error always avenges itself on us later, and 
the risk we run of falling into it is so obvious that all kinds 
of methods are resorted to in order to magnify these slight 
effects and bring them within the range of our perception. 

2. The case (C—a— — E). It is found that on a van- 
ishing in the observation or on its being experimentally 
suspended the whole of E vanishes. In such cases we 
naturally incline to the assumption that a alone is actively 



54 



INDUCTIONS FROM PERCEPTION [Book II. 



concerned in bringing about or at any rate in sustaining E. 
That this may be the case, but is not so universally, we 
learn by comparing this with other observations; let us 
instead of a cause the other parts of C to vanish one after 
the other ; then we shall often find that the whole E dis- 
appears in exactly the same way with the intermission of 
b and d ; from which we gather that it depends not upon a 
single part of C, but upon the simultaneous presence and 
conjunction of all or at any rate of several of them. Every 
complex machine, every living body affords an example of 
this ; in both there are many parts the lesion of any one of 
which is enough to put an end to the motion of the one and 
the life of the other, although no one of them by itself would 
have been able without the co-operation of the rest to pro- 
duce and sustain motion and life. The fact that with the 
destruction of a single part of the brain a a definite spiritual 
function ceases is no proof that just this single part was the 
organ, which produced that function ; even the counter 
experience, that no lesion of other single parts has the same 
result, does not render this conclusion perfectly certain ; it 
always remains possible, that a was no more than the in- 
dispensable part, in which the effect of all the rest took just 
this form E. The function must then cease just as much 
when a is hampered as when all the remaining parts besides 
a of the brain are hampered in the discharge of their 
functions. In order to settle whether it is so or not, we 
must try to observe the changes of E into E 1 , which arise 
when a is left undisturbed, while the functions of the 
remaining organs are checked in their action. 

3. The case (C—a = E + a). The part a disappears in 
our observation from C or is by experimental means made 
to lapse, whereupon the effect E acquires a new element a 
which it had not before, or anyhow an effect a now arises 
for the first time ; we may then conclude that the remaining 
parts of C involve the ground to which a is due, but that a 
hindered that ground from taking effect in such a way that 



Chap. VII.] 



TYPES OF INDUCTION. 



55 



on the removal of a, a can for the first time exhibit itself. 
But the observation does not entirely justify the conclusion ; 
for it remains open to doubt, whether when a disappeared 
a new and hitherto unnoticed condition Z did not enter, 
which alone has to do with the production of a, a being 
capable neither of producing a nor of arresting it. We set 
aside this doubt by an experiment, which makes us sure 
that the means we took to suspend a really produced no 
other or further effect than this negation of and did not 
at the same time contain a positive influence Z, to which 
the appearance of a can be attributed. Whenever a state 
of equilibrium is destroyed by removing one of the forces 
which preserved it, we have an instance of this ; in the 
economy of living functions also Physiology meets with a 
variety of such cases. Suppose the severing of a nerve 
elicits violent movements, and that we can make sure, as in 
this case we can, that the act of severance has not produced 
any lasting and positive excitement, but has only annulled 
an influence which was active before ; in such a case we 
cannot help believing that the organisation is so planned 
that one function holds another in check, and we therefore 
refer the movements observed to the removal of the 
check. There is much in history, which, if we consider it, 
leads us to take the same view. Of course there are 
positive stimuli, which have driven mankind along a certain 
path of historical development, but the majority of great 
and violent revolutions are due to the removal of checks, 
which prevented ever present tendencies and feelings from 
unfolding themselves ; and even those positive impulses for 
the most part only guide events for a while in their particular 
directions ; after a time everything takes another turn, 
because unwittingly and unintentionally the given impulse 
has removed or weakened the checks which restrained forces 
of quite a different kind and of a different tendency. 

4. The case (C—a=-E—a). This case does not require 
considering afresh, but has already been dealt with under 



56 INDUCTIONS FROM PERCEPTION [Book II. 

the second and third heads. If the suspension of a part a 
of C occasions the disappearance of a part a of the effect, a 
and a must necessarily be connected as cause and effect, 
and a may be the exclusive cause of a and a the exclusive 
effect of <z, so that a is the cause adequate to produce or 
sustain a ; but it may be that a is only either the one or the 
other combined with the remaining parts of C, and this last 
may continue to be the case, even when counter experiments 
shew that any other part of C may be suspended without a 
being annulled in the same way, for the parts which still 
remain may serve as substitutes for the parts set aside. 
And this is not all. It may be that a does but indirectly 
condition a, as in the third case ; then another part of C, 
perhaps d+f, is the cause which produces and sustains a, 
only a third part b arrests the action of d +f, while finally 
this check in turn is balanced by a in such a way that the 
removal of a enables the counteracting force of b to suppress 
a. All the other conditions of life C may be left unim- 
paired, yet if only oxygen a be withdrawn, the living 
functions of the animal body are suspended, without so 
visibly altering its structure in other respects as a different 
cause of death might alter it. No one has ever ventured 
to infer from this fact, that oxygen by itself produces life ; 
it was plain that it could only produce life in union with the 
constituent parts of the body, that is, as a stimulus acting 
upon these, or as one collateral cause among many. There 
were some however who ascribed to it a more positive role ; 
it was they declared the very stimulus, which by its direct 
influence awakens and elicits those organic movements, of 
which life is the totality. It has been impossible to confute 
this interpretation of the facts altogether ; but it is certainly 
not the whole truth ; it only shares in the truth along with 
another view according to which the influence of oxygen 
consists in the removal of obstacles, which these functions 
themselves throw in the way of their own further continu- 
ance, owing to their consumption of the organised materials. 



Chap. VII.] 



TYPES OF INDUCTION. 



57 



5. The case (C + a = E). If a fresh cause a be added to 
C without having been before contained therein, the aggre- 
gate effect E can only remain unmodified under certain 
conditions, which are the same as those under which, in the 
first case, the lapse of a constituent part a until then con- 
tained in C left E unaltered. Two cases are possible. 
Either there is to be found in the observation, while a is 
present in it, a Z which escapes our notice although it 
cancels the effect of a, or our experiment is at fault and we 
have not succeeded in introducing a in such a way that it 
can exhibit its effect. If however a is really active the 
aggregate E must be really altered into E\ but this change 
may either withdraw itself from observation or it may not 
affect the particular part of the aggregate E 9 to which in our 
negligence we alone directed our attention, in which case it 
will equally pass unnoticed. 

6. The case (C +a — E + a). A fresh element # , on being 
added to the causes C which have thus far operated, gives 
rise to the fresh element a in what was the effect E. It may 
then be that a is by itself the sufficient cause which would 
produce the effect a in the objects in question. But it may 
also be the case that <z, like the last drop which makes a 
cup overflow, is no more than the cause which completes 
the tale, so that neither those previous causes without it, 
nor it without them, would have brought about this result. 
Lastly, it may happen, that the effect E or more generally 
the fact E, which by being augmented by a passes over into 
E\ is neither a mere state nor an event ever repeating or 
continuing itself in the same way. It may itself be a pro- 
cess of development or growth, which once generated by a 
group of causes C is forthwith constrained by the nature of 
the objects, on which these causes act, to transform itself 
from E into E 1 ; in that case a is an idle addition to C, or 
such an addition as may no doubt have its effect on other 
occasions, but on this has nothing to do with the entrance 
of a. When we introduce into a patient's system the drug 



58 



INDUCTIONS FROM PERCEPTION, [Book II. 



a we are left in doubt, whether the favourable turn a taken 
by the disease is due to the reputed efficacy of the drug, or 
whether the disease would not have taken the same turn 
without a at all. It is not altogether easy to settle such a 
doubt, because in this case the possibility of experiment is 
confined within such narrow limits. If we have once ob- 
served that in several cases the desired result has ensued 
upon the introduction of a, we shrink from the experiment 
of omitting what may be but is not certainly superfluous. 
On the other hand counter experiences may offer them- 
selves unsought and seem to prove that a is not wanted, 
and yet not remove the ambiguity. The cases compared 
are seldom of quite the same kind, and it can hardly be 
proved that the a which is now left out has not found a 
substitute in some Z of equivalent influence. We meet 
with similar embarrassment in dealing with social and his- 
torical phenomena ; it is difficult to decide whether a new 
turn taken by events is to be attributed to a measure or oc- 
currence which is observed to precede it. Still harder is 
it to discover wherein the efficacy of a in all cases consists, 
and what collateral conditions involved in C favour it and 
render it possible. 

7. Thecase (C+a = C+b). The causes have an element 
C in common, but have also distinct elements a and b 
which differentiate them. It is impossible that two such 
causes should have exactly the same effect, but it is quite 
possible that of their aggregate effect a certain portion on 
which we fix our attention is the same, or lastly that so 
much of this portion as falls within our observation is the 
same. Such is the case which I denote by the above for- 
mula. The most obvious conclusion to draw from it is of 
course that both causes produce a like effect in virtue of 
their like element, and that so far as regards this effect their 
dissimilarity is without influence. I hardly need say that 
this conclusion is very often the correct one, even when 
two or more causes have nothing else in common than just 



Chap. VII.] 



TYPES OF INDUCTION. 



59 



a few attributes, while a and b, in which they disagree, dilate 
into clusters of very many attributes. But it may be that C 
by itself alone never produces or sustains the like effect we 
spoke of; in order to do so it may always require to be 
backed up by a or b or in which case we must regard the 
latter elements as equivalent and companion causes of E. 
It may even be the case that the part C, which remains the 
same in different causes, is quite inoperative as regards E 
and that E is entirely dependent on the unlike elements of 
the two. Let three forces act on a particle situate in a 
fixed plane, the one of them C acting along a line vertical 
to the plane, the other two a and b along divergent lines in 
the plane ; we may then quite well substitute for the two 
latter forces others, which give the same resultant. The 
first force d, the only one common to both systems of 
forces, is at the same time the only one which does not 
help to determine the direction and magnitude of the re- 
sultant. It is universally true that any balance of forces and 
any movement admits of being resolved in a thousand ways 
into very different combinations of particular causes. It 
may of course be objected that in all such cases a, b, and d 
are not so widely different from each other as to be dis- 
parate, that they still involve a common element x in spite 
of their differences. This x we shall be told must be rec- 
koned as belonging to the common C, and then C + x will 
always be the true cause of the like effect E. It may be 
answered that such an objection though true is yet irrele- 
vant, for it amounts to no more than a restatement of what 
in the abstract is a truism, viz. that like consequents always 
have like grounds. In this connexion however we are deal- 
ing not with the abstract but with the concrete, and are 
concerned to know the guise in which these like grounds of 
like consequences appear in the actual observation, and we 
found just now that the like elements or attributes in two 
causes are not always the vehicle of these like grounds. On 
the contrary these like grounds are in fact often concealed 



6o INDUCTIONS FROM PERCEPTION. [Book II. 

in the combination of prima facie unlike elements, attri- 
butes, or conditions. These ambiguities then must be got 
rid of by means of collateral experiments. We must know 
whether C alone is able to produce or sustain E ; if it is, 
then of course it does not necessarily follow that a and b 
are without effect, but they are anyhow elements in the 
cause, which we could do without, inasmuch as we then get 
the case (C — a — JE) and its consequences as above con- 
sidered. We must know furthermore if a and b alone pro- 
duce E ox no ; if they do, then agreeably to the same first 
case C is not necessarily inoperative, though it will be no 
more than a contributing cause of E, which might be dis- 
pensed with. If neither the one nor the other is the case, 
then C+a, C+b, C+d are pairs of mutually indispensable 
contributing causes of E, and it is now time, by new com- 
binations of our perceptions or by varying the experiments, 
to find out what is the common element x in a, b, and </, 
and perhaps also what is the particular element c in C, 
which together constitute the true and sufficient cause c+x 
of the identical effect E. 

263. By means of the inferences which we have thus far 
passed in review, we shall not always be able to determine 
even the proximate and sufficient causes of an effect, much 
less to find out the sort of causal tie, which holds the two 
together. To do both is our end and aim and we shall 
come nearer to reaching it, if we can observe the changes 
of quantity on the part of the effects, which attend changes 
on the part of the causes. There is scarcely any sort of 
effect, which does not admit of some quantitative change or 
other. Even such effects as do not directly display a more 
or a less, may be made to do so indirectly. Thus a state of 
equilibrium cannot be more or less equilibrium, but it may 
oppose a greater or less resistance to any attempts made to 
destroy it, or the force needed to maintain it may vary. As 
before, I group together the simplest cases we can observe 
by way of example. 



Chap.VIL] QUANTITATIVE CHANGES. 



61 



i. The case {mE — m C). Let us represent to ourselves 
once more the pure case, which we before denoted by B F 
and may now denote by C E. This formula means that C 
is the whole cause and nothing else than the whole cause of 
E, E the whole effect and nothing but the effect of C. 
Assuming then that both terms admit of direct quantitative 
determination, we regard it as a self-evident principle that 
like differences between two values of C will find a response 
in like differences between corresponding values of E, Cand 
E thus standing in simple direct proportion to one another. 
Then m E will be = m C. This formula is no mathematical 
equation but a logical symbol and presupposes that the 
effect no less than the cause is capable of being measured 
by a standard of its own suited to its nature and permissible 
in its case. It asserts that if this is so the unit of the effect 
E is contained in any effect whatever exactly the same 
number of times as the unit of the cause C is contained in 
the cause which operates to produce that effect. This rela- 
tion however is self-evident only in cases where a number m 
of particular causes C produce, each by itself, the same 
effect E, and where all we do is to add up the sum of these 
separate effects, which will then be proportional to the sum 
of the causes. Suppose we pay away the same amount C 
of money on m different occasions, and each time for the 
same amount of goods E, then assuming that prices remain 
stationary, the total bought will be m E when the total spent 
is m C. Let us take another example. Suppose m to be 
equal but separate impacts which act on the same number 
of different elements and give to each of them the velocity 
E, then the sum of all the velocities produced will be m . E, 
or the amount of the motion generated will be m . E, if we 
regard the number of elements as the index of the mass. It 
is otherwise if the several causes and their effects are actually 
bound up with one another. A lump sum m C will buy 
more goods than would the same sum in m separate pur- 
chases. Here there intervene complicated commercial 



62 INDUCTIONS FROM PERCEPTION. [Book II. 



considerations, which enhance its value in the eyes of the 
seller • in the abstract it remains true that each C is no 
more than the condition and adequate ground of a single 
E ; this is the only consequent which in the abstract the 
ground justifies, it is only in the real effect that it is modified 
by those accessory causes. In the same way an impulse C 
may give a body the velocity E, but m C if it acts on the 
body all at once is not unlikely to shatter it instead of 
moving it forward, m C always remains the rational ground 
of the velocity m E, but the result is modified by other cir- 
cumstances consisting in the texture of the body. There is 
only one condition under which it is self-evident that we can 
expect the cause m C to be followed by its due effect, the 
motion mE y viz. that we may regard a material element as 
the mere substratum of motion, destitute of any native 
power of reaction of its own. We may put it in a general 
way thus. In applying our principle we suppose the m fold 
cause to be equal to m particular causes C, and assume that 
there are no circumstances of any sort present, which would 
compel a single element in this sum to take more or less or 
other effect than if it were present alone and the rest of the 
terms not there. The m fold cause will then produce the m 
fold effect, and conversely in cases where our observations 
reveal this relation approximately we may be pretty sure 
that we have before us a pure case C E, which is identical 
in the sense specified with a pure causal relation B F. 

2. The case (E = Ql). It is often the case that a cause 
C acts on the same object t times, / being understood 
to mean either the number of times this action regarded 
as momentary is repeated, or the number of time units, 
in each of which the continuous force C produces a certain 
degree of effect. Now if this force is of such a kind as to 
allow the object exposed to its influence to remain identi- 
cally the same, the same effect would take place afresh 
in the object every fresh time we let the same cause 
operate on it. Thus after the cause had operated / times 



Chap. VIL] 



QUANTITATIVE CHANGES. 



63 



or after the time t had elapsed there ought to be present 
in the body t times as much effect, provided that is that 
agreeably to the law of persistence every earlier given effect 
is preserved and not annulled by any counteracting force. 
This is the case with motion in space, in the case of which 
we presuppose that the causes producing motion either 
do not change the object moved or only bring about 
in it inner states, which throughout exercise no counter- 
acting influence upon its assumption of new motions e. 
If by the effect E we understand the velocity generated, 
E will always = e . / and depend on the time. Now for 
an opposite case. A constant cause C acts continuously 
on an object during a certain time t, during the whole 
of which time the object maintains an uniform state E, 
always equal to the constant (£. Such a case cannot be 
a pure one ; besides C there must be contributing causes 
or conditions Z, which cancel the influence of the law 
of persistency and render it impossible for the particular 
impulses to accumulate, and thus would leave the effect E 
constant and independent of the time. One more example. 
A cold body grows warm under the rays of the sun, and is 
then found to maintain a constant temperature for any 
length of time during which it is further exposed to the 
same rays. The mere incidence of the rays cannot have 
caused this phenomenon; it is only accountable for on 
the hypothesis of a companion cause, namely the radiation 
which proceeds from the heated body : when it has once 
reached a certain temperature relative to its surroundings 
it is obliged by the law of radiation to give out in its 
turn just so much heat as it continues to receive. 
fit 

3. The case (d E = —d C). There is really no case ex- 
cept that of simple movement through space in which 
we can assume that the effect produced in the object a 
will not in any way prejudice the effect immediately to 
follow. In general this a is changed by the first operation 



64 



INDUCTIONS FROM PERCEPTION. 



[Book II. 



into a : and this fact, that the object that receives the 
effect does not remain the same, constitutes a variable 
concomitant condition Z, which associates with each fresh 
impulse of the cause C effects of which each is more 
different from the first than that which it succeeds. Let 
us first assume that the change of a into a is of such a 
kind as to tend to thwart the next operation of the cause, 
in the same way as an already compressed body offers 
resistance to any fresh compression, as the mutual ap- 
proximation of its elements increases the repulsions 
operative between them. The measure of this resistance 
cannot be a constant quantity independent of all the 
agencies, which here co-operate. It must on the one 
hand be proportional to the specific intensity of the inner 
repulsions, to which the resistance is due and which are 
different for different bodies ; on the other hand it must be 
proportional to the amount of compression already effected, 
since it is this which by bringing the elements closer to 
one another in the manner described intensifies their mutual 
repulsions. In the former of these two conditions we get 
a constant coefficient for the influence which the cause 
C may still exert, a coefficient which depends on the nature 
of the object a ; the other condition compels the amount 
of such a subsequent influence to stand in inverse ratio 
to the amount of the result E already attained to, and 
this last amount itself continues to depend in case of two 
different causes C and C n on their respective amounts. 
Now natural causes are never quite instantaneous in their 
action. We can analyse every C into a number of d C, 
which are successive, though for the rest their distribution 
in time is arbitrary. Each of these fractions d C of the 
cause would if it acted singly produce a corresponding 
and constant fraction of the effect dE — m . d C, but 
inasmuch as each of them acts on an object which is 
already modified by the action of its predecessor, the effect 
dE is altered for each of them. It therefore makes no 



Chap. VII.] QUANTITATIVE CHANGES. 



6~ 



difference whether we regard C and C n as two different 
causes or as two different values, at which one and the 
same growing cause C has stopped in its growth or is 
for the purpose of our analysis supposed to have stopped. 
If we then signify by E=f(C n ) the result already produced 
by n successive d C, we obtain for the effect dE which 
will result from the addition of yet another dC the fol- 

771 

lowing : dE= — * d C. Among pure quantitative functions 
E 

it is the logarithm C, which shows this mode of growth, 
and so we come upon logarithmic expressions in calculating 
operations which by their own results create obstacles to 
their own repetition proportional to those results. 

4. The case (dE = m E d C). We have just seen that 
a cause cannot when repeated have its effect diminished 
merely because it is not acting for the first time. Just 
as little can its effect be increased by the mere fact that 
it has already acted several times. Both effects are ascribed 
to habituation : we say ' practice makes perfect,' and also 
' habit hardens.' An increase in the effect produced obliges 
us, no less than a decrease, to assume a contributing cause 
Z; this Z consists in such a modification of the object 
a influenced by the cause into a as facilitates every sub- 
sequent operation of that cause by continually opposing 
to it less and less resistance. Thus the first blow shakes 
a stone in such a way that the second blow has only to 
intensify the vibrations already going on within it in order 
to overcome the cohesion of its parts. If nothing else 
enters into the calculation we must for reasons of the 
same sort as in the above case reckon the magnitude of 
the effect produced as at any moment proportional to 
the aggregate result or to the integral of the earlier effects. 
In the case of pure quantitative functions of C it is the 
exponential function e% which 1 possesses this peculiar 

1 [There is here an unavoidable ambiguity of notation, e which was 
before the symbol for effect, here stands for the base of Napierian 
logarithms.] 

Logic, Vol. II. F 



66 



INDUCTIONS FROM PERCEPTION. 



property of a differential quotient equal to the integral 
itself. Thus we shall often meet with applications of 
this formula as well as of the other in mathematical ex- 
pressions of the forms which natural effects assume. 

5. The case (d E = m . sin C). In no one of the cases 
which we have thus far examined do we get effects, which 
alternately increase and decrease at the same time that 
their causes go on steadily increasing. Whenever therefore 
E periodically alternates from increase to decrease, while 
C changes in one uniform direction, there must exist 
besides C one or more companion causes Z, the relations 
of which to C are either in themselves variable or are 
so deranged because it happens that they operate together 
that the effects of all now accumulate upon and now 
cancel each other, and so pass through maxima and minima 
from the one to the other of these reciprocal attitudes. 
We can conceive of the combinations possible in this 
case being infinitely numerous ; the formula I have used 
is no more than a very inadequate symbolical expression of 
these possibilities. 



CHAPTER VIII. 

The Discovery of Laws. 

264. In the relations between causes and effects examined 
in the previous chapter lie the clues by which we are guided 
in instituting fresh experiments or seeking for fresh ob- 
servations in order to exclude the possibility, which still 
remains, that different causes may produce the same effect. 
The general import of this procedure is always the same : 
from the impure observations SP or C E we have to 
discover the pure case 2 II or B by eliminating from the 
observation all that has nothing to do with the causal nexus 
before us. I see no reason to analyse this general precept 
any further into a number of separate methods. It is much 
more worth our while to point out that in elementary 
algebra we have already an instructive type of the very 
various modes of operation by which we may reach this 
end. We have given us equations which jointly determine 
the relations of two or more unknown quantities ; these 
equations we transform in all sorts of ways by adding on 
new quantities, by subtracting others, by multiplication and 
division of the whole ; and are thus at last able to compare 
the equations immediately with one another, and adapt 
them to the elimination of particular unknown quantities. 
The present problem is to be solved in a similar way, now 
by a timely addition of fresh conditions, whose influence we 
can calculate ; now by a suspension, equally calculable in 
effect, of given conditions ; or again by altering the relative 
position of the co-operating cause ; or, lastly, by modifying 

F 2 



68 



THE DISCOVERY OF LAWS. [Book II. 



our own attitude towards the material we have to observe. 
I will not stop to decide whether we shall ever be able to 
reach by such means a pure case B E; but even supposing 
we were so lucky as to have discovered the exact cause C 
of the exact effect E, we should yet in no case have com- 
pletely satisfied our curiosity, save in the case of historical 
enquiry. For the only conclusion we could draw from this 
pure case C E would be that whenever the same C really 
recurred the same E must attend it. But the practical 
needs of life, no less than the interests of science, urge 
upon us the further question : how will E change into E l 
when C passes over into C 1 , or what shape will an effect E 
have to assume when the place of the C observed is taken 
by another C 1 , of which we can state exactly how it differs 
from C ? In a word, we desire not only to be certain that 
there really is a connexion between C and E, but to know 
the law according to which that connexion comes about 
and varies. 

265. The term law has different meanings as we use it 
in connexion with different circles of human interests. Its 
logical meaning however never varies. Stated in its com- 
plete logical form a law is always a universal hypothetical 
judgment, which states that whenever C is or holds good, 
E is or holds good, and that, whenever C undergoes a 
definite change into C 1 through a variation of itself d C, E 
also becomes E l through a definite variation of itself d E 
which depends on d C. A law is hypothetical, because it is 
never meant to be a mere enumeration of what happens ; 
its sole function is to determine what should or must happen 
when certain conditions are given. All laws are thus hypo- 
thetical in their import, and those which refer to permanently 
given or permanently presupposed conditions are no excep- 
tion to the rule ; they only seem to be so because they are 
not stated in the form of an hypothesis. Thus we enunciate 
the following in a categorical form as a law of nature : all 
ponderable elements attract one another inversely as the 



Chap. VIII.] LA W NOT MERE GENERAL FACT. 



69 



square of their distances from each other. Here we merely 
state the fact that in the case of such elements a particular 
condition is adequate to produce this consequence; this 
condition is perpetually fulfilled and consists in their simul- 
taneous presence in the same world. Again the constitution 
of a state categorically maps out the relations which hold 
between the various groups of its members, but always 
under the tacit proviso that so long as the state exists at 
all, these fixed ordinances shall be constantly maintained 
and renewed as generation succeeds to generation. 

But besides being hypothetical — that it is as a matter of 
course — every law is also universal, and must on that account 
be as strictly distinguished from a mere universal matter-of- 
fact as from a decree made for a particular case. Kepler's 
law that all planets move in ellipses round the sun, which is 
fixed at one of the foci, is originally no law at all, but the 
mere expression of a fact. It gets the name of law, thanks 
to the accessory idea (which is perfectly justified) that all 
planets owe their movement to a common ground, and that 
we may therefore assume that the proposition will continue 
to hold good no longer as a mere proposition, but as an 
actual law for bodies which are still beyond our ken, pro- 
vided always that they show themselves to be planets by 
revolving round the sun. A law which gives powers of 
expropriation for the purpose of laying down a particular 
line of railway is logically considered a decree or mandate ; 
but inasmuch as the mandate is not arbitrarily given but is 
based on a general law, which pronounces expropriation 
under certain conditions to be always legitimate, it may 
fairly itself assume the more pretentious name. It is implied 
in the idea of a law that it should pay such regard to 
variations or differences in the condition and consequent, 
only the idea cannot always be realised. The certainty 
that two bodies attract each other is in itself a fact which 
needs to be further determined; natural science does not 
see in it a law until it can assign the particular ratio in 



7o THE DISCOVERY OF LAWS. [Book II. 



which the attractive force varies in its amount in depen- 
dence on differences and variations in the mass and 
distance of the bodies, or on some other condition of 
variable magnitude. 

It is the same with moral and judicial laws also. A com- 
mandment so universal as that which enjoins love of our 
neighbours may fairly, as an expression of the deepest 
motive which can govern us, possess a higher value than 
can any law, yet in its form it lacks the precision of a law ; 
for it is neither clear prima facie what result should follow 
from such love, nor in actual life can the commandment be 
fulfilled, without the love which it prescribes — whatever it 
may consist in — having a definite degree of liveliness, or 
without its force flowing along a channel in one case along 
which it does not flow in another. The general formula we 
have quoted gives no hint whatever as to what this channel 
shall be. Judicial laws, on the other hand, are based on 
the distributive suum cuique in its widest significance. 
Whether they prescribe actions or fix penalties, the predi- 
cate they attach to every case of the recurrence of what 
they bring under the general notion of any legal relation is 
not intended to be incapable of modification. Differences 
in quantity between various cases have a real significance 
in the eye of the law : it is only the defectiveness of our 
standards for determining those differences, which compels 
us in practice to be content with roughly graduating the 
scale of legal consequences, when we would far rather make 
it exactly proportionate to the individual differences on 
which those consequences depend. It would seem that 
none but purely negative laws and moral prohibitions 
ignore any such graduation of ground and consequent. 
I leave it to the reader however to judge, whether in a 
theoretical sphere negative judgments are to be regarded as 
laws at all, and not rather as contrapositions, in which for 
merely logical purposes we have changed the positive as- 
sertion of a law into the negation of its opposite. In any 



Chap. VIII.] 



LAW AND RULE. 



case by putting it in the form of a universal negative, we 
lose a part of the truth, viz. the measure of the distance by 
which each case is separated from the predicate, which is 
simply denied of them all As regards moral prohibitions, 
it is true that we do not find in them as such any reference 
to such a gradation or adjustment of penalty to guilt, never- 
theless in passing judgment on a breach of them, we always 
make such reference. They prohibit beforehand any appro- 
priation of another's property, but the commission of such 
an act is according to its particular nature subjected to very 
various degrees of disapprobation and punishment. 

266. There is a difference of intention between a law 
and a rule, which may in most cases be easily seized, though 
it cannot be maintained in all. In practical life a law 
determines a state which is to be brought about by an 
activity or mode of conduct, and which is essential to the 
fulfilment of the ends of the political or social community ; 
the rule supervenes as a practical ordinance, and since there 
are many possible courses of conduct, all in themselves 
equally contributory to the realisation of that state, the rule 
helps us partly to select the most advantageous of those 
courses, partly to secure, if only by fixing a definite mode 
of procedure, the requisite uniformity and harmony between 
individual performances. In theoretical investigations of 
reality, we mean by a law the expression of the peculiar 
inward relation which exists between two facts and con- 
stitutes the ground at once of their conjunction and of the 
manner of this conjunction ; and in every simple case there 
is but one law. The rule, on the other hand, prescribes a 
number of logical or mathematical operations of thought, by 
which we are so to combine our perceptions as to arrive at 
conclusions, which in their turn tally with reality, and there 
may be several such rules all equally sound for one and the 
same case. Thus it is only for the law that we claim an 
objective truth. The rule is merely subjective, and sums up 
the various adjustments of our thought, by which, starting 



72 



THE DISCOVERY OF LAWS. 



[Book II. 



from the standpoint we occupy over against things, we so 
far master their connexion as to be able to calculate and 
predict the consequences flowing from given facts of reality, 
and divine aright their antecedent grounds and causes. 
These operations of thought which the rule prescribes need 
not take the same path as the development of things them- 
selves. They need not necessarily move a principio ad 
principiatum ; instead of the conditions on which a thing 
really depends they may employ trustworthy signs or 
symptoms. They must never indeed lose all connexion 
with the reality, but they are free to make use of any round- 
about method, which our attitude towards things necessi- 
tates, and to transform the inner relations of things as they 
like. This difference in intention between a law and a rule 
is no doubt a wide one, yet in making it we are hardly ever 
quite unbeset by doubts, least of all where we are concerned 
with the investigation of reality. It is clear at the outset 
that not a few of the methods of procedure at present in 
vogue are mere rules ; but more than that, it remains an 
open question whether any one of the laws, which we 
believe ourselves to have discovered, really deserves the name 
in the special sense explained above. We are accustomed to 
use the name where we have reached very simple and univer- 
sal propositions about the actual conjunction of phenomena. 
Thus we regard it not as a rule, but as a law of nature, that 
the force of gravitation diminishes according to the square 
of the distance ; yet the inner nexus between the terms of 
this proposition is still undiscovered, and we do not know 
how it is that the quantity of space, which intervenes 
between two bodies, can cause their reciprocal effect to vary 
as it does. Ultimately, therefore, even this law is a mere 
rule, which teaches us how to calculate from given data of 
distance and mass the variations of their effects ; it does not 
exhibit the inner connexion of these effects with their con- 
ditions. We shall have occasion to recur later on to this 
question. At present it is enough to notice that in the 




Chap. VIII.] 



PERSONAL ERROR. 



73 



considerations which immediately follow we shall look on 
the law as no more than the simplest rule which conjecture 
has to guide it in getting at the genuine nature of things. 

267. Thus far we suppose that the means specified have 
enabled us to discover as accurately as possible the pure 
causal connexion between C and E. We also suppose our 
experiments or observations to have supplied us with a 
number of pairs of values of this cause and its correspond- 
ing effect, these values being quantitatively determined and 
forming a double series. It is anticipating somewhat, yet 
we may suitably preface our attempt to determine the uni- 
versal law of such a double series by a consideration of the 
various causes, which may produce a divergence between 
the quantitative relations which we find and the true rela- 
tions of things which we are in quest of. In the first place 
let it be borne in mind that what we observe is not the 
things themselves, but the impressions, which things make 
upon us. We will not at present attempt to settle whether 
the impression produced in our consciousness can ever be 
like the things and their relations which produce it. One 
thing however is clear on the face of it, and that is, that it 
is not obliged to be like them, but may change with every 
change in the disposition of the recipient subject. Hence 
a doubt as to how far we can conclude from the subjective 
excitements produced in us by an assumed external world 
to the objective nature of this reality, and this doubt affects 
the whole realm of our knowledge. We will not go into it 
at present, but are content to understand by such truth or 
correctness of our observations, as we at present aspire to, 
their universal validity for all human observers, who are 
normally constituted and placed under similar conditions. 
If it is asked how we can be sure that any particular obser- 
vation possesses such universality, we can only answer that 
practically the ultimate decision in every case rests with the 
overwhelming majority who agree in their views, as opposed 
to the minority who disagree. If anything appears to me 



74 



THE DISCOVERY OF LAWS. [Book IL 



different from what it appears to everyone else under exactly 
similar conditions, there must be some error in my indi- 
vidual observation, an error which will vary and may be set 
right by repeating the observation, if it can be traced to 
mere momentary inattention, but which becomes a perma- 
nent, and in a narrower sense personal error, when the 
anomalous organisation of the individual's senses is to blame 
for it. How widespread is such defectiveness of sensible 
apprehension in regard to the qualitative content of sensa- 
tion, is shown by the way people will differ in their judg- 
ments as to the resemblance and contrast of colors, or the 
harmony and dissonance of tones. Such disagreements, 
however, are equally noticeable where it is a quantity which 
has to be estimated. For all practical determinations of 
quantities given in reality ultimately rest on the accuracy 
of our sensible impressions, and all that artificial methods 
and instruments of measurement can do is to transform 
what is too big or too small, the one by splitting it up, the 
other by somehow magnifying it, in such a way as to bring 
both within the sphere of more moderate intermediate 
quantities, of whose equality or inequality we can judge with 
sufficient accuracy by help of our sensible faculties alone. 
And really it is to such a simple judgment as this last that 
all our measurements are reducible. Nature does not 
endow us with a power of specifying offhand how great is 
the difference between two unequal quantities of space, or 
time, or intensity ; we only acquire such a power by long 
practice, and then very imperfectly. All that we are directly 
sensible of is that two quantities of the same kind are on 
the whole equal or unequal ; the amount of their difference 
is measured in an indirect way by finding out how many 
definite and equal units of quantity taken together exactly 
make up that difference. We say that a line b is bigger 
than a line because to begin with it contains a length 
equal to a, while perception reveals to us a further residue d, 
which that a does not contain. The size of d is only to be 



Chap, vill.] RELATIVITY OF MEASURE. 



75 



found by employing a standard of length, and it is found 
the more accurately the smaller the units, which we can 
distinctly observe by our senses, and which added all 
together produce a length equal to d. But even if we use a 
microscopical standard we must admit that everything 
ultimately depends on the certainty with which sense-per- 
ception shews us that the extremity of the d to be measured 
exactly coincides with the extremity of one of these infini- 
tesimal units of measurement. When intervals of time are 
equal we recognise them pretty accurately as being so in 
virtue of our feeling of the equality of one beat with another; 
but we can only measure unequal intervals against each 
other by dividing them into beats or equal recurring 
units ; nothing but the immediate sensible impression, 
however, informs us of the equality of these units them- 
selves. And when we use clockwork to mark the re- 
currence of these units with audible ticks, the accuracy with 
which it does this still rests ultimately on the certainty and 
precision of the visual impressions, which helped us to set 
out the spatial dimensions of the works and their parts in 
such a way, that their movement shall in fact give out those 
signals at equal intervals. Lastly, if this expedient is to 
serve to fix the times, on the expiration of which certain 
phenomena only observable by other senses, as by the eye, 
occur, nothing but the immediate impression can tell us 
that a phenomenon of this other kind exactly coincides in 
time with this audible signal, and it is just here as we know 
that our judgment is for physiological reasons not so acute 
as we could wish it to be — on the contrary, it needs the 
previous correction of our personal error. 

In conclusion, I will but briefly mention, what is familiar 
to every one, the relativity of all our determinations of 
measure. There is nothing absolute except the numerals 
by the help of which we count the recurrences and specify 
the number of units found. The units themselves can only 
be determined relatively to each other, and there is no sense 



76 



THE DISCOVERY OF LAWS. 



[Book II. 



in asking how big anything is unless we measure it by a 
presupposed standard. To find those units, that is, to 
determine them in such a way that they may be fixed, 
useful, and unambiguous, is itself a problem, which the art 
of observation has to solve. It is enough at present to 
remark, that in unchangeable natural bodies we have a 
means of determining units of length, while we have exact 
periodic astronomical appearances whereby to determine 
units of time ; and if it is the intensity of moving forces that 
we have to measure, we can sometimes observe how they 
balance each other, sometimes what velocities they generate. 
As yet however we are without means of arriving at observ- 
able units of measurement for the strength of sensations, 
feelings, and desires. 

268. Supposing that this primary defect, the personal 
error, has been remedied, what we observe may still fall 
very far short of the truth, owing to the position, which 
either individually or as men generally we occupy towards 
things themselves. We could illustrate this from other than 
spatial phenomena, still it is they which enable us most 
readily to appreciate the frequency with which the same 
process or the same object yields very different images 
according to the point of view of the spectator. I think I 
may hazard the assertion that every regular event gives a 
regular projection of itself for any point of view we like to 
take, but the rules by which we reason from one such phase 
of the object to another are framed in such a way as to favor 
one point of view more than another, and on that account 
it is often exceedingly difficult to go back from the event as 
projected to the event which produces the projection. A cir- 
cular movement will only appear circular to a spectator, 
whose standpoint is somewhere in the line drawn through 
the centre of the circle at right angles to its plane \ to an 
eye situated anywhere outside this axis and this plane it will 
appear an oval ; while if one views it from any point in the 
plane of the circle but outside its circumference, it will appear 



Chap. VIII.] ERROR DUE TO POSITION. 



77 



as an oscillation in a straight line. The synthesis of the 
times traversed by the moving point and the loci corre- 
sponding to the times will form a separate series for each 
point of view, and each such series will be regular in its 
formation, though one of them will have much more value 
than another as an indication of what really takes place. 
Now if this was all that met our observation and if we had 
not already got a stock of other experiences in regard to 
what is true in reality and of usual occurrence, we should 
have no reason to desire any other rule than that, say, which 
in our example, expresses the rectilinear oscillation. But in 
nature we are seldom left without secondary features, which 
force themselves simultaneously on our observation and 
lead us first to doubt and then to correct our first impres- 
sions. That we observe that circular movement means not 
that we think or represent it mentally but that we see it, and 
we only see it if rays of light are reflected from it on our 
eyes. Hence it follows that changes in the apparent size 
and illumination of the body must accompany its movement 
for every observer who is placed outside the axis. Only a 
person who takes his standpoint in that axis itself can fail 
to notice these variations and so feel no impulse to seek an 
explanation of them. Now let us place ourselves in the very 
plane of the circle, the body will then, as it travels from one 
extremity a of its apparently rectilinear path towards the 
middle of the same, wax in size and brightness, while after 
passing the middle it will wane in both respects till it reaches 
b • if it then recedes from b to a this decrease in bulk and 
brightness continues at first, reaches its minimum at mid- 
path, while from these onwards to a the body waxes afresh. 
If one takes it that all these appearances are real, one has 
many questions to answer. Why does the body reverse the 
direction in which it is moving when it reaches the extremi- 
ties of its path, and why does its velocity increase as it 
approaches the middle and decrease as it approaches the 
ends ? Either there is something in that middle point the 



78 THE DISCOVERY OF LAWS, [Book II. 



effect of which is to draw the body towards it, or there must 
be present and at work in the prolongations of its path equal 
and opposed forces urging it in that direction. But why, if 
that be so, should it pass through both the minimum and 
the maximum of its size and brightness at the same middle 
point and yet the force or forces remain uniform? The 
easiest conjecture to make would be that the two appear- 
ances were merely coincident ; quite apart it might be said 
from its movement along its path the body is subject to 
periodic increases and decreases of bulk, which however 
are merely functions of the time not of the place. Still as 
at any time / it must be in some place or other it may at 
the moment of its greatest bulk just as well be in the middle 
of its path as anywhere else, and as its bulk requires in 
order to reach its minimum the time which it takes to ac- 
complish a half oscillation, this minimum too must take 
place just as the body occupies this same middle point. 

But who would credit such an explanation as that ? In 
the rest of nature such periodic enlargements are altogether 
unheard of, while changes, such as we have described, in 
the apparent size and brightness of bodies are quite familiar; 
we know that bodies are liable to them according as they 
alter their distance from our eye. Relying on such analogies 
then we shall try to grasp or apprehend the fact observed as 
the projection of other and truer facts. We notice no with- 
drawal of the body between the loci of the maximum and 
minimum, on the contrary both coincide with each other in 
the middle of the path. Moreover the ways by which it 
goes and returns appear coincident at every point. Taking 
all these considerations together we are obliged to suppose 
that the true path is a plane closed-up curve, one of the dia- 
meters of which must lie along our line of sight at the centre 
of its apparent path. By comparing the particular apparent 
loci occupied at successive moments of time we shall further 
discover whether the true path is a circle, an ellipse, an 
oval, or what. The mere mention of the name of Coper- 



Chap. VIII.] INTERPRETATION OF SERIES. 



79 



nicus will be enough to make the reader understand how 
the accumulation of insoluble difficulties in the facts as 
observed impels us to transform our views of nature, and 
how much at once becomes clear when we grasp what is 
sensibly given as a mere projection of a reality beyond our 
observation. In order to that however we must already 
be in possession of a store of universal truths as well as of 
earlier experiences of facts; pure logical precepts may stimu- 
late but cannot conduct us to the goal. 

269. We must now go back a step. Before we try to 
interpret the observed facts in the manner specified, we 
must be in possession of the actual laws, which we think of 
reducing by means of such interpretation to a form at once 
simpler and more in correspondence with the real course of 
things. Nothing is given to help us in the discovery of 
these laws beyond the series of values displayed by the 
causes and their corresponding effects. Now even if we 
assume that these numbers before us are perfectly correct 
as a statement of what we succeeded in observing, still the 
transition from this series of isolated terms to the universal 
law of its formation is always a jump on the part of thought. 
How do we know that such and such a law is the only one 
valid for the series and true ? There is no process of 
demonstration by which we can find such a law, none by 
which it could be shown to be what it claims to be. We 
can never do more than guess at the law and then by the 
help of innumerable secondary considerations heighten the 
probability of its being the true one. It is of importance to 
be quite clear on this point. If we have to start with a 
limited number, say n terms of a numerical series given in 
the order in which they succeed one another in the series, 
it will be easy to find a simple general formula, exactly 
corresponding to these given n terms and expressing their 
general term ; but even then this formula need not 
necessarily be the only possible one : it may at least be 
apprehended in different ways. For example, let the given 



8c 



THE DISCOVERY OF LAWS. [Book II. 



terms be i, 3, 5, 7, 9 ; then if 1 denotes the place in the 
series occupied by the first of the given terms, 2^ — 1 will 
exactly express the general term. But if we think the 
general term in exactly this manner it will hardly correspond 
to a real physical relation, of which it is meant to serve as 
the regular expression. The same given series may however 
be thought as an arithmetical progression with the initial 
term 1 and the difference 2, and besides that as the series 
of differences got by subtracting the square of a whole 
number from the square of the one following it in the 
numerical series. Both readings of the series may be 
expressed by the same general term, both determine every 
term of this series, but the genesis of each term is con- 
ceived differently in the one case and in the other, and 
this difference of manner is of importance, because it 
allows of our making different assumptions in regard to 
the physical relations of the phenomena expressed by this 
series. 

Thus without going any further we here have unsolved 
doubts in plenty. But this is not all. The presuppositions 
we make in this case are not at all the same as what we 
make in the case of observations ; a general term found in 
the manner just described holds exactly for only the n 
terms, from which it has been generalised. Not so with 
the laws which must be generalised from observations ; we 
require these to hold good no less for the values of the 
causes and effects which we have not observed than for 
those which we have. We can of course interpolate terms 
in a given series ; that is, we can calculate missing links in 
such a way that they will fit into a series agreeably to a law 
of its formation, which we have beforehand abstracted from 
the given terms, and which often proves to be not a little 
complex. But then we assume that the particular law 
developed from the given terms holds equally good for 
terms not given, — an assumption which is always permissible 
when we are merely concerned with the completion of a 



Chap. VIII.] RISKS OF REGULAR OBSERVATIONS, 8 1 



conceivable series, but which is altogether inadmissible when 
the question is whether this conceivable series itself corre- 
sponds to a something real even where this correspondence 
has not been observed. Thus before we demand that a 
law, which we have somehow got out of the given terms, 
should be extended to terms not given, we must have 
reasons, which justify our pursuing such a method of inter- 
polation at all. We may illustrate this by a very simple 
example. Let us figure to ourselves the values of C as so 
many abcissae „r, each larger than the last by A x, and the 
values of E as so many ordinates y. Now if the given series 
gives the same value y — B for all values m A x of x, it may 
of course be the case that the equation would hold good for 
all the unobserved ordinates, which correspond to fractions 
of a A x. In that case the line joining the extremities of all 
the ordinates is a straight line and parallel to the axis in 
which the abscissae are taken. Still this does not follow as 
of course. Take any two A x we like, the ordinate y between 
their extremities may have every possible value, and the 
curve which unites the various co-ordinates y may describe 
every conceivable path. It may be real or imaginary, 
straight or crooked, y may pass through one or several 
maxima or minima, even through infinity, and all these 
indeterminable paths may be as different as you please 
in the interval of one A x from what they are in that of 
another. From such considerations as these we may derive 
a minor rule for selecting observations, like that we have 
noticed above for imperfect inductions. The rule in ques- 
tion forbids us to form the series of pairs of values in such 
a way that C progresses according to a regular law, and 
none but the particular values of E are permitted to appear, 
which correspond to these symmetrical values of C. If we 
do so the chances are that we shall only get a series of 
singular values, of maxima or minima, or fixed values of E y 
which periodically recur, and which either give us no 
insight or suggest false surmises as to the intermediate 

Logic, Vol. II. G 



82 



THE DISCOVERY OF LAWS. [Book II. 



course of the curve. A regular advance of C by equal 
increments no doubt helps us to guess the universal law 
of the series ; but if we wish to confirm this guess we must 
make the increments of C change as unsymmetrically and 
irregularly as possible. To put it quite simply, a man who 
never observes a place of public resort but once in every 
seven days and that on a Sunday afternoon, has no right to 
suppose because it is crowded then, that it is as crowded 
on a week-day. A man who never looks at the moon but 
through a chink which only allows him to see it at its full 
height, cannot guess the path it pursues through the heavens 
for the rest of its time. If on the other hand we find that 
the values which y assumes for intermediate values of x, 
taken at random from between the values already con- 
sidered, adapt themselves to the law derived from these 
latter, we have for the first time some justification for inter- 
polating all the other y's in conformity with this law. 
Strict logic would not admit even this to be a complete 
justification ; so long as it is impossible to observe all the 
successive values of C and all the corresponding effects £, 
so long we remain in doubt whether the law which holds 
good in the cases observed would hold in those not 
observed. 

This doubt is narrowed in practice by collateral consi- 
derations based not on general principles of logic, but on 
our actual knowledge, which as a rule is enough for the 
purpose, — of the matter under investigation. If for instance 
we are investigating the way in which a particular natural 
force acts, we know for certain that E cannot be infinite 
for any finite value of C ; and we shall know enough of the 
peculiar character of the force in question to be able to 
judge whether it is possible for its effects to increase steadily, 
or to oscillate periodically, or to sink to zero for particular 
values of C; lastly we shall know if they are likely to 
accumulate undiminished by the lapse of time, or whether 
we must assume that some counterforce is constantly an- 



Chap. VIII.] FACTS CANNOT GIVE COMPLETE LAW. 83 

nulling wholly or partially the results generated. It is these 
assumptions, which are grounded in fact, which justify our 
transferring the law for the pairs of values we have actually 
observed to values not observed, and of doing so with a 
strong probability of being right. There is still another 
expedient in cases, where there is no restriction to the 
number of possible expedients. By means of autographic 
arrangements attached to the apparatus in which the effects 
of the force are rendered visible, we can compel the force 
to register of itself the results which it produces at each 
moment of its continuous working. By help of such 
mechanical means our observations, which would otherwise 
be always limited in number, are so infinitely extended that 
they follow each other without any break, and the visible 
curve thus generated allows us to form as safe a judgment 
as can possibly be based on observation in regard to the 
continued or intermittent nature of the effect, the uniformity, 
retardation, or acceleration of its rate and its periodical or 
non-periodical increase. It is always of course open to 
those who are given to logical hyper-criticism to object that 
every curve drawn consists ultimately of a series of point-like 
deposits of pigment and that these only appear as a con- 
tinuous line to the naked eye, which interpolates whether 
we like it or no. After all, we may be told, you have only 
got a number of particular perceptions and these do not 
allow you to infer the nature of effects, which found no 
pigment available to register themselves by and which 
therefore correspond to the gaps between the colored 
points, which make up the curve. Let us leave such 
objections to answer themselves; all I wish to do is to 
accentuate the truth that the discovery of an universal law 
is always a guess on the part of the imagination, made 
possible by a knowledge of facts. This knowledge is 
recalled to our memory by the resemblance of the given 
case to analogous earlier cases, and thus offers itself as an 
explanation. But a demonstrative method, or a method 

g 2 



8 4 



THE DISCOVERY OF LAWS. 



[Book II. 



which involves no logical jumps, a sure logical receipt for 
arriving at the true universal law of a series of events, does 
not exist. 

270. If we return to our series of values in order to see 
how far the problem in hand is successfully solved, we are 
confronted by numerous cases in which it emphatically is 
not. Among such are all those statistical calculations, 
which view a result jS, which really depends on the co- 
operation of several conditions, by sole reference to the 
influence of a single one of these conditions, and then 
attempt to find an universal law in regard to the relations 
of the two. Thus it is attempted to estimate a man's 
present expectation of life by sole reference to the age he 
has already reached. This self-contradictoriness of the 
problem at once shows itself ; if a variable quantity E is 
a function of C, x, y, and z, we cannot express it as a mere 
function of C alone, entirely neglecting x, y, z } which ought 
to enter into the true expression as part of the collateral 
conditions. Nor in fact would a man ever make such an 
attempt unless, once more, he had experience which taught 
him to put some trust in it. However much the procedure 
may lack precision from a theoretical point of view, he still 
knows that as a matter of fact something comes of it, though 
not quite what he wishes ; and conversely it is the absence 
of all result in other cases, which induces him to abstain 
from similar attempts. What result we do usually arrive at 
in such cases is based on the following considerations. 
Among the conditions on which the continuance of a man's 
life depends, that which in estimating it we regard as the 
most important is beyond doubt the age C, which he has 
already attained ; for inseparably bound up with that age is 
a modification of his bodily system, which continues slowly 
to run its course and is ultimately sufficient of itself, even 
though all other conditions remain favorable, to make 
death inevitable. During long periods of one's life however 
the action of C changes slowly and inconsiderably, while 



Chap. VIII.] DISTURBING CONDITIONS. 



85 



in other sections of one's life it increases very quickly and 
significantly; hence it follows that the same outward con- 
ditions have an uniform influence on the body during one 
period of life, and during another an equally uniform but 
uniformly different influence. It is upon this interaction of 
the present stock of vitality and circumstances that a man's 
capacity of further life really depends, and so we may 
suppose that between certain fixed ages the expectation of 
life decreases according to one tolerably constant law, 
between certain other limited ages according to a different 
but equally constant law ; we cannot however conceive of 
an universal law which should determine the expectation of 
life universally for the whole of life, and so for any age a 
man may have reached. In such investigations therefore 
partial laws or formulas are usually laid down, which are 
only meant to hold each of them for values of C which lie 
between two fixed limits, and to help us to estimate the 
-corresponding values of E. Theoretical significance these 
formulae have none ; they are merely practical short cuts 
or synoptical expressions of how things take place in the 
gross ; if they are very simple and yet exact enough for our 
purpose, they aid our calculations ; if they are, to start with, 
of a complicated nature it is at best empty affectation to 
lay them down at all ; in such cases it is more useful to go 
back to the original form of a table containing in its simplest 
form the mass of observed facts, from which they were 
derived. 

271. But matters may be less unfavorable, and we may be 
able to reckon on the presence of a universal law capable of 
being expressed by the help of two centres of relation C and 
E. The question then arises, which we are to choose of the 
many laws that may with equal truth or with equal approxima- 
tion to the truth be supposed to underlie the series of pairs 
of values presented us. In raising this question we make 
assumptions slightly different from those we have hitherto 
made. The numerical terms of our series will not repre- 



80 



THE DISCOVERY OF LAWS. [Book tC 



sent the observable facts with such complete accuracy as 
we supposed before ; they will contain inaccuracies, but we 
are for the present content to believe that these are small, 
and that they are not all on one side, but exceed the truth 
about as often as they fall short of it. Accepting these 
conditions a doubt arises whether the particular formula, 
which fits in most accurately with the given values, is to be 
regarded at all as the law we want. The pure case B F 
will hardly fall within our observation quite unalloyed ; the 
result which the condition B would by itself alone involve 
will be somewhat altered by the simultaneous co-operation 
of other causes which' we can never wholly eliminate, and 
this matter of fact, impure already, will be still further 
modified for the worse by the slight flaws which are insepar- 
able from our observation. Thus the data from which we 
start involve what we want along with disturbing elements 
which we do not want ; a formula, which was exactly ad- 
justed to those data, would be a copy of this mixed matter 
of fact, but not a law for the pure case, which we sought to 
separate from its alloy of accidental and irrelevant circum- 
stances. This consideration forms the general ground 
upon which we permit ourselves if at all to neglect the 
slight divergences, which still remain between the given 
values and a law approximately covering them ; we then 
put down these differences as due to unknown disturbing 
causes. Cases however may arise, in which a law com- 
pletely answers to the given values and must yet be regarded 
as not the true one, or anyhow as less true than another, 
which answers less closely to them ; this will be the case 
when there are known disturbing causes, which must 
necessarily act, but of which we find no hint given in the 
former law. Let us assume that two bodies a and b revolve 
together on different planes and at different distances round 
a third c, which steadily attracts both ; and that it strictly 
follows from our observations, that the two bodies describe 
two similar regular ellipses : then either the observation 



Chap. VIII.] SIMPLEX SIGILLUM VERI? 



87 



itself must be pronounced defective, or the elliptical orbit 
cannot be regarded as the law of these movements in the 
desired sense. For if we only admit first the attraction 
between c and a and between c and b and admit of none 
between a and b, a fortiori however if we do admit there to 
be attraction between these two, the path, which a would 
describe, were b not present, must be modified when b is 
present together with it. Either therefore the real paths of 
the two bodies, when they are moving together, diverge 
from a true ellipse, in which case our observations are 
inaccurate and fail to reveal to us these slight divergences ; 
or the ellipse is the actual path of either body, in which 
case the path prescribed by law is some other one, which 
they would traverse, except for these disturbances. For 
after all in such investigations as these our aim is not 
merely to get an universal expression or copy of the facts 
as they result from the application of an universal law to 
the definite conditions of a special case; what we do want 
is rather such a general statement of the law as will allow 
us, just because these special circumstances are excluded, 
to judge of the results which would follow, though the 
collateral circumstances were changed, from the same or 
similar main conditions. In such cases then as this we 
shall be inclined to doubt the truth of an assumed law, 
when it fits in with a faultless and all too striking exactness 
to the given observations. If it be asked what other law 
should be held to be a truer one, we answer that that can 
only be conjectured according as the disturbances we dis- 
regarded can be estimated on other grounds. The doubt 
raised in us however may induce us to combine our ob- 
servations in a new way, or to institute experiments which 
may throw light on the matter. 

272. In case there are several laws, which all come about 
equally near to fitting the data before us, we are accustomed 
conformably to the above to prefer the simpler and to see 
in simplicity as it were a guarantee of truth. Against this 



SS 



THE DISCOVERY OF LAWS. 



[Book II. 



view, which raises the simplex sigillum veri into a universal 
principle, logic must enter a no less universal protest. If 
what we have to do is to calculate a special case by the 
light of a general law, the simpler formula is of course to 
be preferred, because it is more convenient) but from a 
more general point of view its simplicity is no test of its 
truth or probability. We must carefully consider what we 
may generally expect in the particular field, which we would 
explore. If it is clear that in that field a result E depends 
on divers independent determining elements, then a simple 
law expressive of their connexion is of course not im- 
possible, but extremely unlikely. Properly the first feeling 
we should have on finding such a law would be one of 
distrust in its validity ; we should believe we had taken 
things too easily in our observations or in our reasonings 
and had left out of sight some of the essential conditions ; 
we shall only be satisfied if a searching investigation shows 
that these neglected conditions really always cancel one 
another's influence in such a way as to justify our excluding 
any reference to them in the universal law. Say we have 
found from mere observation that a body starting from the 
surface of a sphere under the attraction of the centre of 
the sphere always reaches a certain other concentric surface 
with the same final velocity, no matter along what path 
it passes from the one surface to the other ; such a remark- 
able discovery as this we could only credit on one condition, 
namely if it were shown that this remarkable compensation 
of different collateral conditions really takes and must take 
place in the case. 

We are easily deceived in similar cases, when the 
result found is not so paradoxical as the above. The 



formula T= tt 




seems to unite all the determining 



elements, upon which the time of a pendulum's swing 
depends, for a superficial observation does not give any 
effect to the angle of vibration. A more exact theory how- 



Chap. VIII.] 



SIMPLEX SIG1LLVM VERI? 



89 



ever shows that this simple expression is only approximative 
and that the true law is far more complicated. A certain 
speculative principle which we may come across later on 
leads us to suppose beforehand that in nature there exists 
a variety of compensatory arrangements, in virtue of which 
certain types of resulting events are maintained in perpetual 
conformity with the same simple law, no matter how dif- 
ferent the medium, through which in particular cases these 
types are realised. Nevertheless one should only count 
upon the presence of such arrangements, where observation 
beyond doubt reveals them ; on the other hand, where we 
have no means of thus forecasting the limits, within which 
the result of imperfectly known conditions must confine 
itself, the supposition of simple laws and the predilection 
for them remains an error only to be guarded against by 
a thorough exploration of all essential details of the given 
object of investigation. The present state of natural science 
does not perhaps make these warnings so necessary as they 
were a score of years ago, when there was a strong tendency 
to explain such complex phenomena as organic life by very 
simple, but no less inadequate principles. Of course it 
is very different when the object with which we are dealing 
belongs to a class of phenomena, which we cannot regard 
as changeable products of a number of independent causes, 
but rather as themselves manifestations of those funda- 
mental forces, whose constant action under all sorts of 
secondary conditions makes up the complex tissue of 
physical processes. For these cases which do in fact 
realise approximately or completely the type of the pre- 
supposed pure case B F, we certainly must regard the 
simplicity of the law as a sign of its probable validity; 
yet not for the somewhat aesthetic reason that simplicity 
is in all cases a characteristic of truth, but because for 
these pure cases only one of the simple forms of regular 
coherence — already noticed (§ 263) — between cause and 
effect is in fact conceivable. 



90 THE DISCOVERY OF LAWS. [Book II. 

273. The reader will have noticed how much importance 
attaches to already acquired knowledge for the discovery 
of new laws, and how we even went so far as to appeal 
to all manner of previous considerations and accessory 
thoughts, through which alone the immediate data of sense 
come to have a precise meaning. The usual way of stating 
the necessity we are thus under is to say that we need 
hypotheses, in order to make use of the results of obser- 
vation. In fact we may be inclined to apply this term 
hypothesis to several of the modes of thought, of which 
we have already availed ourselves. Thus we may say it 
was to make an hypothesis to infer back as we did from a 
periodical increase and decrease of an effect while the cause 
constantly increases to a shifting of the relative positions 
of the active elements associated in the cause. I think fit 
however in the interests of logic to define terms differently 
and to distinguish between postulates, hypotheses, and fictions. 
The regressive inference just mentioned is a postulate, that 
is, it expresses the conditions which must be set up, or the 
ground of explanation which must be given by some reality, 
whether thing, force, or event, before we can think the 
phenomenon in the form in which it is presented to us ; 
it thus requires or postulates the presence of something 
that can account for the given effect. The postulate is not 
therefore an assumption which we can indifferently make 
or leave alone, or discard for another; rather it is an 
absolutely necessary assumption, without which the content 
of the observation with which we are dealing would contra- 
dict the laws of our thought. Nor is it at all necessary that 
the postulate should be so indefinite in respect of its content 
as it might appear to be, judging by the way in which I have 
just described it ; on the contrary, what must be there, or 
have been there, or be accomplished in order that we may 
conceive the given phenomenon as really happening may 
be something altogether definite. What is left indefinite is 
the answer to be given to an essentially different question, 



Chap. VIII. ] POSTULATE AND HYPOTHESIS, 



0 1 



namely, the question who or what that is, which by its con- 
crete nature introduces exactly those conditions, which ac- 
cording to the postulate must needs be fulfilled in order that 
the given appearance may be possible. If a body of known 
mass moves in a known curvilinear path with a known 
velocity, we can assign with perfect accuracy the sum of 
the conditions, i. e. of the resultants B, B l . . . which must 
act upon the body at every moment if it is to move in this 
way. All that remains indefinite is the source from which 
B and B 1 come, whether they are both of them simple 
impacts of simple forces or themselves the resultants of 
several joint forces, whether in short they are effects of 
forces or communications of already existing movements. 
It is clearly an abuse of language at once to apply the term 
hypothesis to all sucn demands of thought. If someone 
merely tells us that this curvilinear path requires forces of 
a certain intensity and direction to divert the motion from 
the tangent just so much in every moment, we should 
answer him in some such way as this : you teach us nothing 
which we did not know before, you merely repeat the 
conditions, which it was evident from a bare analysis of the 
given appearance must be supplied by any theory, which 
could be brought forward in explanation of the facts. 

But we mean something else by an hypothesis ; we mean 
by it a conjecture, which seeks to fill up the postulate thus 
abstractly stated by specifying the concrete causes, forces, 
or processes, out of which the given phenomenon really 
arose in this particular case, while in other cases may be 
the same postulate is to be satisfied by utterly different 
though equivalent combinations of forces or active ele- 
ments. Thus we may fix at once two characteristics of 
the hypothesis. Firstly it is far from being an empty 
surmise, which comes into our heads without any reason 
at all : it always rests on a postulate, which we must accept, 
and is designed to explain the contradictions or lacunae, 
which make the given appearance prima facie unthinkable. 



Q2 



THE DISCOVERY OF LAWS. 



[Book II. 



These it explains by assuming a secret inner organisation 
of real things and processes, in which these contradictions 
vanish, while at the same time it becomes conceivable how 
and why the said contradictions unavoidably arise for us 
in the outward appearance, which alone we can observe. 
The second characteristic of an hypothesis is closely con- 
nected with the former : every hypothesis is meant to be 
an account of a fact, and is no mere figure of thought or 
means of envisaging the object. A person who sets up an 
hypothesis believes he has extended the series of real facts 
which he can observe by a happy divination of facts not less 
real though falling outside the range of his observation. In 
such a case there is no need for the facts thus divined to be 
simple and ultimate facts ; they in their turn may give rise 
to researches going still further back into the grounds of 
their possibility; it is enough for the hypothesis if the facts 
it supposes can be conceived as really existing, though we 
reserve for another time the question how they come into 
existence. Students of Optics found (to put it briefly) that 
observed facts make it necessary to postulate that rays of 
light act in the same moment in a different manner on their 
right side and on their left, and that this action itself 
alternates incessantly with the time, and that therefore there 
must be some cause capable of bringing about just this 
phenomenon. The physical hypothesis was that this pos- 
tulate would be satisfied by transverse vibrations on the 
part of atoms of ether. What may be the source of these 
transverse vibrations, which form so indispensable a pre- 
liminary in the explanation of the phenomena, remains a 
question for the future to solve; in any case however it 
involves no contradiction, which would prevent our con- 
ceiving it as a process which actually takes place. 

We have still to explain what we mean by fictions. These 
are assumptions made by us with full consciousness of the 
impossibility of the thing assumed, whether it be because it 
is self-contradictory or because for other than intrinsic 



Chap. VIII.] 



FICTION. 



93 



reasons it cannot pass muster as a constituent of reality. 
Fictions are of use when there is no proposition T, under 
which a given case M can be logically subsumed as a case 
of its application, whereas there is a proposition T 1 , from 
the actual applications of which M is only distinguished by 
a definite difference d. We then class M under T\ draw 
therefrom the conclusions we want and correct them later 
on by adding on such modifications 6, as are rendered 
necessary by the distinction d, which still remains. The 
finding of the circumference of the circle by inclusion of it 
between an outer and an inner polygon may be regarded as 
merely a method of limitation, unless we like to consider 
the conception of the length of a curve as in itself a sort of 
fiction ; but anyhow the formula d s 2 = d x 2 + dy 2 is certainly 
a fiction, if the symbol — signifies real equality and not the 
mere approximation thereto. As long as d s is a real arc, so 
long the equation is false ; but as soon as ds loses all quan- 
tity all the terms become nought and the equation loses all 
meaning. It leads however to an infinite approximation to 
the true value, as by gradually diminishing ds we gradually 
diminish the error committed and by so doing render the 
sum or the integral of ds ultimately independent of it. It 
is hardly requisite to remark upon the extraordinary import- 
ance of such modes of procedure for the intellectual process 
of discovery; but we also encounter them in other branches 
of knowledge, and the lawyers' custom of turning to the 
most nearly allied maxim of law T\ when there is no 
special rule under which a particular case may be brought, 
is from a logical point of view to be classed as a fiction, 
though we generally apply the name only to cases of a 
peculiar kind. Jurisprudence must of course be left to 
shape its own nomenclature; still I cannot believe that 
what used to be regarded as a fiction was not something 
more than a mere transfer determined by a fresh act of 
legislation of all a man's legal rights and obligations to a 
subject, who per se stood in no relation to these. I think it 



94 



THE DISCOVERY OF LAWS. 



[Book II. 



depended on something further, and in the case of the 
Roman adoption the assumption of the father's name who 
adopted seems to me to prove that, as a psychological fact, 
an attempt was made to begin with to regard a relationship 
which could not be established in reality as yet after all 
established, while the corresponding sum of rights and duties 
was determined as the result and on the basis of this function. 

274. So important are the results which we expect from 
hypothesis that we cannot blame the attempt so often made 
to subject to some sort of discipline the free course of the 
discoverer's imagination from which alone hypotheses can 
flow. But we must observe that though most of the rules 
laid down are truly excellent so far as they can be carried 
out, yet we must not regard a particular hypothesis as ille- 
gitimate because it disregards them ; if we do we seriously 
curtail the utility of hypothesis. Let us illustrate our mean- 
ing. It is alleged in the first place that the hypothesis must 
satisfy the postulate, on which it is based, not by a fictitious 
representation, but by assigning a reality, and that it should 
therefore make assumption only of what may be thought as 
fact, not of what is inherently self-contradictory. This is 
obvious enough ; still we go too far, if we require that the 
content of an hypothesis should always carry with it the 
possibility of being directly refuted by subsequent observa- 
tion. We may look on this requirement as constituting an 
ideal, and it certainly is a very useful rule to observe, for it 
teaches us, where we can, to construct our hypothesis in 
such a manner, that its falsity, if it be false, transpires at 
once instead of being for ever proof against direct refutation 
by reason of its content being wholly inaccessible to ob- 
servation. Still we should have to sacrifice many useful 
assumptions, if we pressed this demand in all cases. The 
teaching that the points of light, so conspicuous in the 
heavens at night, are bodies of vast size, only very remote 
from us, is at the bottom only an hypothesis, by means of 
which we try to understand the otherwise inexplicable daily 



Chap. VIII.] 



LEGITIMATE HYPOTHESIS, 



95 



and yearly motions of these lights. However false this 
assumption may be, it is clear that no future advance of 
science can ever directly refute it ; we must therefore abide 
content if our hypotheses are thinkable and useful, if they 
are capable of explaining all interconnected appearances, 
even such as were still unknown when we constructed them, 
if that is to say they are indirectly confirmed by the agree- 
ment of all that can be deduced from them in thought with 
the actual progress of experience. But if we would be so 
fortunate as to find an hypothesis, which will not lack 
this subsequent confirmation, we must not simply assume 
anything that can be barely conceived as real; we must 
only assume that, which besides being thinkable conforms 
so to speak to the universal customs of reality, or to the 
special local customs which prevail in that department of 
phenomena to which the object we are investigating belongs. 
We do proceed so in all fields of enquiry. For instance, if 
in the text of a legal enactment a particular phrase only 
admits of an ambiguous deduction being made in regard to 
a given case, we do not interpret it in an arbitrary fashion 
by simply allowing our wits to play freely on it ; we go back 
to the ratio legis on which the formula is based, and by the 
light of that seek to interpret the phrase in a manner suitable 
to the particular case. It is the same in the natural 
sciences ; there too a successful hypothesis is always due to 
the attention paid to analogies noticeable in the material 
world at large or in particular departments thereof. Nothing 
but the analogy of fluids and of the atmosphere could have 
originally suggested the hypothesis of the continuous filling 
of space by matter ; there was nothing in solid bodies to 
suggest the idea, for most of them are not only divisible 
into parts but are composed of a number of actual parts. 
In the case of such bodies the notion of the continuity 
of matter was only applicable in regard to their minute 
parts, and so it became a scientific truth that they consisted 
of discrete atoms, each of which could just fill its own small 



9 o 



THE DISCOVERY OF LAWS. [Book II. 



space continuously and no more. Now when it was found 
that solid bodies became fluid and fluid ones solid, and 
that even gases assume solid and liquid shapes under 
certain conditions, the atomistic theory was fully justified 
from that point of view; it only transferred what was actually 
true of one part of the body or of certain forms of it, to 
other bodies or other forms, in the case of which the same 
state could not as a matter of fact be demonstrated to be 
real, though it could be shown to be possible, inasmuch as 
upon this assumption the appearances presented by them 
remained perfectly conceivable. As soon as it is found that 
certain groups of phenomena are readily explained on the 
supposition that nature habitually acts in such and such a 
way, fresh discoveries are made every day, because people 
at once try how far other facts may be referred to the same 
principle. Such was the case with the undulatory theory. 
On the surface of water, in strings, on resonant surfaces, 
waves could be directly seen and their shape rendered 
visible in particular cases by artificial means ; there was no 
apparent reason for supposing these movements to be 
confined to certain materials and there was accordingly 
much to be said in favour of the hypothesis, which sought 
to explain on the same principle first the propagation of 
sound by the air, next the movement of the luminiferous 
ether, and lastly the phenomena of heat. 

Similarly, in the organic world, people stumbled at a few 
points on a division of labour of which they had never 
dreamed; where before very different functions had been 
attributed to the same substratum, each of these functions 
was shown to have a special organ of its own, which did not 
do service for any other function. It was then suggested 
hypothetically that the same thing went on in regard to the 
nerves organic to the different sensations of colour and 
sound ; whether the truth of the matter has been reached is 
still open to doubt, but from the point of view of logic 
there can be no doubt that the hypothesis is justified. 



Chap. VIII.] SUGGESTION OF HYPOTHESIS. 



91 



Again, movements are often observed in plants, even con- 
tractile movements ; still it does not appear that these are 
due to the contractions of living contractile tissues as are 
the movements of an animal's body; consequently, plausible 
as this hypothesis is in itself, it is not advanced in this case, 
because prima facie it does not seem to agree with the habits 
of nature in this region ; on the other hand it is worth while 
enquiring whether this semblance is not a fallacious one, 

275. There is yet another condition which a hypothesis 
must fulfil. It should be exactly adjusted to the postulate 
which it is framed to meet, and not contain either more or 
less than it must contain if it is to answer to the demands 
of the postulate. Hence a rule which must be carefully 
observed in constructing hypotheses. When we have to 
account for something which happens, we must not look 
vaguely about as if for inspiration ; we must before anything 
else rigorously analyse what is given, and so lay down the 
exact postulate which the hypothesis must satisfy. When 
we have done so we may neglect for the moment the 
secondary features, of which we know from other sources 
that they can easily be treated as mere accessories, when we 
come to define in a more concrete manner any hypothesis 
which can come under discussion ; but all essential elements 
of the problem, all, that is, which are not themselves mere 
consequences of other elements, must be accurately observed, 
for it is entirely from the way in which they are conjoined 
that we have to conjecture the most suitable form of the 
hypothesis we shall choose. We must then make a survey 
of our world, to see if it contains any elements, causes, 
forces, or combinations of forces of such a kind as to satisfy 
the postulate laid down ; after the fullest survey of these 
and guided at once by a practical and theoretical motive we 
shall make choice of those which fulfil the specified require- 
ments in the simplest manner and in the most complete 
accord with the ruling analogies of the particular department 
in question. 

Logic, Vol. II. H 



98 



THE DISCOVER Y OF LA WS. [Book IL 



For example, a body is found covered with wounds ; our 
first concern will be to settle whether the wounds must 
have been inflicted while the man was still alive or after he 
was dead ; we then try to estimate the magnitude, mode, 
and direction of the forces, which could have caused the 
wounds. Having thus ascertained the conditions we found 
a postulate on them, and enquire whether this postulate is 
satisfied by assuming a mere natural force to have acted or 
only by presupposing a weapon to have been wilfully 
employed. This enquiry may be said to settle the form of 
the hypothesis, e.g. we may have to make the assumption 
that murder was committed, after which we proceed to 
detect the agent not by the help of ill-founded fancies, but 
by asking ourselves what persons there are of whom the 
deed might be expected, partly because their relations to 
the murdered man would have supplied a motive, partly 
because there is nothing in their characters to prevent our 
suspecting them without direct evidence. We have no 
space to give in all the necessary detail an example, which 
would illustrate the extreme care which in judicial investi- 
gation is taken to satisfy every part of the postulate; a 
conviction founded on it is not regarded as a safe one, 
unless it accounts for every single circumstance, which 
because it is a violation of the ordinary course of things 
would require to be specially accounted for even if we were 
not dealing with a case of felony. In such a case a man is 
forced to be circumspect by the vastness of the issues at 
stake ; his judgment is rendered keener by the thought, and 
he reasons with far greater accuracy than he would in con- 
ducting many a philosophical speculation, in which much 
worse errors are condoned because they can do no one any 
harm. We find plenty of people who without seriously 
examining some phenomenon which strikes them as strange 
put it down to what they call the fluid of animal-magnetism, 
and this without specifying the circumstances which need 
to be explained. They talk in a vague and general way of 



Chap. VIII.] THE 'LEAST CAUSE? 



90 



this fluid being emitted and immitted, forgetting that such 
barren generalities are perfectly useless as explanations of 
the kind, quantity, and sequence of the phenomena for 
which they are supposed to account. Natural science is not 
so liable to go wrong in this way because it must state its 
problems with so much mathematical precision even to 
render them intelligible. 

276. I shall presently have to speak of individual facts, 
in treating of which the important point is not so much the 
simplicity of the hypothesis framed with a view to their com- 
prehension as the completeness with which the hypothesis 
covers all that is contained in the facts. Experience teaches 
us in how many roundabout ways an event is sometimes 
brought about in a particular case, whereas in other cases it 
may arise from several simpler causes. But at present we 
are not concerned with individual facts ; we are still trying 
to discover the matter-of-fact which is the common basis of 
a whole class of frequently recurring events ; and here in 
deference to a sort of principle of ' the least cause ' we must 
prefer the simpler hypothesis to the more complicated : not 
because simplicity in itself is any guarantee of truth, but 
because if we go out of our way to assume any datum what- 
ever, which is not indispensable in order to the production 
of the thing, we make an utterly void supposition over- 
stepping the given postulate and therefore unjustifiable in 
point of method. But our procedure may be logically 
correct without being endorsed by reality. Suppose we have 
selected our hypothesis and are trying to deduce from it the 
original appearance, we may find that our deductions do not 
agree with the data, either because our analysis of the latter 
was defective, or because fresh observations, which were 
impossible before, have brought out new aspects of the 
thing. In that case the hypothesis must needs be amended. 
This may be done in two ways. The hypothesis contains 
elements, which in themselves admit of being modified, and 
we determine these in a more suitable manner, so that as 

H 2 



IOO 



THE DISCOVERY OF LAWS. 



[Book II. 



grounds for the deduction of the given fact they are no 
longer either too wide or too narrow but just adequate. 
The other way is to add on fresh subsidiary hypotheses in 
regard to a few of its leading features. In advocating this 
mode of procedure just here I am at issue with a much- 
advocated theory which regards such a grafting of fresh 
hypotheses upon old ones as a sufficient proof of the 
inadmissibility of the latter, and insists that we ought to 
replace them at once by simpler ones. We do not really 
act upon any such theory either in everyday life or in 
science. We do not pull a house down and build it up 
anew just to get rid of a flaw, which a slight modification of 
its construction would remedy; we do not at once devise a 
brand new constitution when a few provisions of the old one 
begin to be oppressive ; and widespread as is unfortunately 
the tendency to ride principles to death, the opportune 
adjustment of necessary changes to what is permanently 
good in old institutions has always been considered the true 
art of statesmanship. And if we look at the way in which 
the body of science has grown up historically we see that it 
too is very willing to essay new points of view under old 
and incommodious forms, if only not to lose any of the 
truths which have once been won through those forms. I 
do not mean to say that science should or will rest content 
with such methods ; we all trust that the result of all our 
painful investigations may prove a simple and thoroughly 
consistent whole ; but until we have arrived at that result 
we must not be deterred by the oddly complex and patch- 
work garb, in which our views must needs be clothed, if we 
are careful to adjust them to each freshly known or better 
known feature of our object by means of subsidiary 
hypotheses tacked on to our earlier assumptions in regard 
to it. This is the only way in which we can hope to reach 
the simple and plain result we seek. The more carefully 
we now proceed the more surely may we expect that in the 
course of our procedure (just as in any intricate calculation, 



Chap. VIII.] HYPOTHESIS MUST BE IMAGINABLE. ioi 



which must yield a simple result in a foreseen manner) our 
manifold assumptions will spontaneously reduce themselves 
to simpler and more universal ones, so that in spite of all 
the circuitous reasonings employed a net result will remain 
with us, which is not only simple and synoptical, but com- 
pletely covers every part of our postulate. In conclusion 
all will admit that a lucky gift of insight may make us able 
to do without all these roundabout methods; but logic 
cannot impart inspiration ; the only method it can teach is 
what we have cited : — we must curb our impatience and 
steadily go on transforming a hypothesis once essayed, until 
we educe from its inappropriate transitional forms a simple 
shape of it, which satisfies both our requirements and those 
of the object. We must not be in a hurry to lay down 
before our labour is finished principles good for nothing 
but parade, or we shall be misled into making light of 
problems, into neglecting inconvenient peculiarities, into 
acquiescence in views which in a rough and coarse manner 
reflect the large outlines of a thing, but are quite inadequate 
to account for its particular features. 

277. A nice point remains to be noticed. Nothing can 
seem more imperatively necessary than that a hypothesis, 
which is meant as a conjecture of something which really is 
or happens, must before all things allege nothing but what 
is in itself possible : and of course it must assume nothing 
which is ascertained to be impossible : but still there is a 
doubt as to where the possibility which is still admissible 
begins and ends. I have tried to solve the doubt by care- 
fully choosing my words ; I have said that the hypothesis 
may legitimately involve anything that can be mentally re- 
presented as given matter-of-fact, but nothing else, and I 
really believe not only that this is all we should require but 
that we may admit so much as this to be possible without 
coming into conflict with the idea of the hypothesis. The 
hypothesis intends to conjecture a fact, but it is also content 
that this fact should when conjectured just exist in the way 



102 



THE DISCOVERY OF LAWS. 



[Book II. 



in which facts really observed so often exist : viz. that while 
we can conceive it or picture it we cannot explain the 
manner in which it may possibly come to be. Nothing can 
warrant our assuming by way of hypothesis a circle, which 
is at the same time a triangle ; it is beyond the constructive 
power of our fancy to frame a mental image of such a 
figure, nor could it ever present itself to our senses as a 
given matter of fact. On the other hand we may assume 
the existence of invisibly small yet extended atoms of un- 
changeable shape and size ; there is no contradiction in the 
notion of them, which would prevent our conceiving them 
as possible objects of perception, were our senses rendered 
more acute by artificial means. There is no reason why we 
should not look on the existence of such atoms as real, and 
suppose that though they are inaccessible to our unaided 
senses they are yet the basis of the phenomena which we 
can observe. We may probably have to modify this idea, 
when we try to think it out and examine its possibility as an 
element in the system of nature ; but still there is no need 
to do so, till we have availed ourselves of it as a preliminary 
principle and found it of permanent use in accounting for 
particular phenomena. In the same way the theory of 
transverse undulations of a luminiferous ether answers, we 
saw, the requirements laid down by a postulate of observa- 
tion, and such undulations can no doubt be conceived as 
really taking place, though no light has yet been thrown on 
their physical origin. The entire supposition of an infinitely 
extended homogeneous or isotropic ether is indispensable 
so far as we can see to our theory of the propagation of 
light, but it belongs to the same class of ideas ; we can pic- 
ture it clearly enough, but we cannot in the least see how so 
uniform a distribution of interacting elements is possible as 
a mechanical result. Those who admire the logical methods 
of natural science occasionally deceive themselves, when 
they represent the whole structure of our cognitions as 
resting on absolutely sure foundations ; we are rather like 



Chap. VIII.] 



BASES OF SCIENCE. 



103 



men who are tubbing a well with masonry; like them we 
build from above downwards and so are we obliged to 
assume a substructure of hypothetical facts, which we trust 
will be sufficiently firmly upheld for a time by the unanalysed 
ground at the bottom to support our superstructure, until 
we can carry our knowledge a step deeper down and replace 
the hypothetical basis of our knowledge by a basis of facts, 
and then go through the whole process over again. It 
must be admitted that at this rate we leave a doubt as to 
where hypotheses and fictions, laws and rules respectively 
begin and end ; I have hinted at this idea before and I 
shall recur to it again. 



CHAPTER IX. 

Determination of Individual Facts. 

278. We cannot be certain about a matter of fact unless 
we have ourselves directly perceived it ; and even then only 
on the supposition that our interpretation of the sensible 
impression, which is all that is originally given, is correct. 
We interpret this by combining it in the form of a judgment 
into a whole of interconnected parts. When our information 
comes to us through others, we can only be sure that our 
information is trustworthy when we can rely on the witnesses 
or reporters. There may be much to recommend and justify 
the confidence we repose in them, but nothing can ever 
demonstrate its necessity. Again we habitually argue back 
from given facts to facts not given, but only attested by the 
former as their causes. Every such inference is liable to be 
wrong, because although every consequent must have a 
ground and a single ground adequate to produce it, still 
there may have been several different but equivalent matters 
of fact, all equally entitled to be called the cause — because 
each of them involved the ground — of the given effect, 
Again we frequently argue forwards from observed circum- 
stances or events to a future or contemporaneous fact, which 
however withdraws itself from our observation. There is an 
uncertainty about all such inferences, because every con- 
dition may in the actual course of nature meet with a 
counter condition, which though it never annuls the conse- 
quences of the other, yet hinders them from actually 
assuming the particular form in which, except for that 



RISK OF SIMPLE EXPLANATIONS. 1 05 

hindrance, they would have manifested themselves. It 
follows that wherever anything is outside the range of 
immediate perception, we are in our judgments of reality 
limited to probabilities, and have to look about for means 
by which to raise these probabilities as nearly up to the 
level of certainty as is sufficient for our purpose. 

279. In thinking about such matters we are swayed by 
two very general and somewhat antagonistic principles. In 
the first place there is no such thing as a train of events, 
causally related and belonging to one another, which runs 
its course by itself in a world of its own ; on the contrary, 
every such series of events goes on in one and the same 
world at one and the same time with numberless others. 
It always therefore seems utterly unlikely that any cause 
should unfold without a hitch the whole endless series of 
effects, which would have flowed from it if it could have 
acted alone upon the component parts of the world. A 
conviction that such is the case colours our daily life and 
conversation ; it finds expression in the old warning not to 
moor one's ship by a single anchor, nor one's life by a 
single hope. If we are anxious to bring about a particular 
result we take a variety of precautions, each of which will 
effect what we want ; if one miscarries, another will reach 
it; if they all come to nothing through the operation of 
external disturbing causes, we shall be able to console 
ourselves with the conviction that such a conspiracy of 
chance as would prevent a single one of the many causes 
on which we relied from producing the desired effect was 
quite as improbable as that they should one and all have 
succeeded. In the same way we distrust a historian who 
deduces mighty revolutions from mere trifles, or the doc- 
trinaire who because some tendency really had a decisive 
influence in an earlier epoch, pretends to see in all the 
details of the history of centuries just its reactions and no 
more. The former loses sight of the innumerable collateral 
conditions, in virtue of which alone so trifling an event 



1 06 DETERMINATION OF INDIVIDUAL FACTS. [Book II. 

could even seem to be fraught with such vast consequences ; 
nor do the reasonings of the other carry conviction with 
them; mankind is a collection of many heads, which for 
ever teem with unconnected and incalculable impulses. 
We cannot believe that these have been wholly without 
influence in determining the course of history, especially 
when conjoined with the influences of nature, which follow 
an arbitrary order or disorder of their own. We are 
aesthetically dissatisfied with any poetry which sets before 
us a human character which is unswervingly self-consistent 
in all its actions, great and small ; such a character lacks 
the air of being a genuine creation of reality, because no 
trifling irrationalities of behaviour are ascribed to it, no 
venial but wayward likes or dislikes : such a mere per- 
sonification of an abstract quality is wearisome in fiction, 
while in life, if such a man could live, he would be so 
repulsive that we should hardly feel towards so impersonal 
a being the moral obligations, which are only intelligible 
between persons. No less incredible would a story be in 
which all the endeavours and resolves of a thoughtful man 
were brought to nought by a constant recurrence of adverse 
accidents. Were such a work meant seriously it would 
shock us, and we could only endure it as a bit of comedy 
which awoke in us the soothing reflexion that the whole 
sphere of action was an insignificant one, as well as a happy 
disbelief in the reality of what was being tricked out before 
us as a possibility. Even music seems, not untrue indeed, 
but insipid and unmeaning, if the flow of its melody can be 
too easily discerned beforehand. It must not make the 
simple forward movement which answers to its initial strain; 
it must reveal its living elasticity by the suddenness of the 
turns which it seems constrained to make by obstacles, 
which encounter and thwart it. Lastly we distrust any 
practical project which instead of co-ordinating side by side, 
paratactically, to use a phrase of syntax, independent con- 
ditions of success, lets them depend hypotactically on a web 



Chap. IX.] AND OF COMPLICATED ONES. 107 



of mutually conditioning presuppositions. Such schemes 
only provoke ill-success ; for in multiplying the parts of the 
structure we only multiply the points of contact with hostile 
influences, and by making one depend on another per- 
petuate the effects of a check once received. 

280. The second of the two principles mentioned is 
suggested by the fact, that although we can imagine several 
different groups of equivalent causes agreeing in the attribute 
of producing one particular effect, still each of these groups 
will have in addition to the common effect other and 
peculiar accessory effects of its own which will differentiate 
it from the rest. Now what we look upon as a single matter 
of fact is very often a complex whole composed of manifold 
effects all gathered into one. The different combinations 
of causes then adequate to produce just this complex effect 
will be very few, so much so that may-be only one of the 
many combinations we are accustomed to meet with in our 
experience will be really adequate. So long therefore as a 
given matter-of-fact is only known to us in its large outlines 
we are accustomed to suppose very various causes in order 
to account for it ; as soon however as the finer side traits 
which characterise it come to be known, our choice of 
causes narrows itself considerably, till at last we find that 
there are very few facts, of which we can make hypothesis, 
which will satisfy all the requirements of the postulate 
founded on these data. Among these facts we then select 
that one in particular which at once is the simplest and 
presupposes the least number of mutually independent and 
co-operating elements. Nor is the above principle a ruling 
thought in science only ; it governs the most various con- 
siderations : a whole chain say of simple facts is set before 
us in evidence, which taken in connexion with each other 
may be conveniently explained if we assume that a par- 
ticular deed was committed ; from such a hypothesis we 
can, we will suppose, deduce everything in the facts except 
those slight accessory circumstances, which depend on 



io8 DETERMINA TION OF INDIVIDUAL FACTS. [Book II. 

accidental conditions and really give to each particular 
commission of a deed a peculiar complexion of its own, 
which it shares with none other. The defendant will 
ascribe each link in this chain of incriminating evidence to 
a separate cause compatible with his innocence, and will 
try to explain away the conjunction of them all as due to a 
mere unfortunate coincidence ; but the persons trying him 
will turn their attention exclusively to the assumption which 
explains them all in their connexion with each other, and 
are hardly likely to listen to his forced pleadings. Just in 
the same way a patient often consoles himself by referring 
each of the several symptoms of his malady to a trifling 
cause of its own ; but he does not for all that deceive the 
physician, who by his diagnosis pitilessly exposes the serious 
complaint, which at once renders the concurrence of all 
these accidents conceivable. 

I hardly need add that these obvious principles of judg- 
ment only suffice to recommend one preliminary conjecture 
in preference to another ; where we have important issues 
to decide we must never forget that what is improbable is 
still possible. It is not enough therefore merely to follow 
out to its conclusions the particular assumption which the 
evidence before us forces upon us as the most natural. For 
it even to come near to deserving belief, it is not enough 
that all the evidence should of itself converge in favour of it ; 
we must have carefully tested the less likely suppositions 
which the nature of the matter admits of, and have found 
that they leave just as many lacunae and contradictions in 
the facts to be explained as does the former. Besides this 
we must take care as far as possible to argue only from 
positive evidence ; negative evidence is ambiguous : whether 
it alleges the omission of an action, or the absence of a 
state, it can only be used to prove a matter-of-fact when 
what it denies may be regarded as being necessary under 
any other presupposition. All that follows from a denial of 
anything is just the denial in turn of what we cannot think 



Chap. IX.] 



PROBABILITIES. 



109 



without virtually affirming the thing denied. Lastly in 
deciding a question the mere quantity of evidence matters 
little — what matters is the quantity of independent evidences. 
And in this connexion we must be on our guard against a 
common form of error in reasoning. We may be right in 
punishing a fault once, but when its inevitable consequences 
crop up again and again the inclination we feel to chastise 
the offender over again for each of these in turn is wrong : 
in the same way the probability of a conjecture is unfairly 
exaggerated for us, when after the mark which first led us 
to make it, the consequences necessarily involved in the 
possession of that mark gradually disclose themselves ; no 
doubt they agree with our conjecture, but we cannot use 
them to strengthen it. In conclusion, the observance of all 
these rules, of whose application it would be much more 
interesting to give examples than to formulate them in this 
dry logical manner, is compatible with much error ; still we 
must not underrate their real value on that account. One 
practical maxim we may draw from the consideration of all 
these imperfections : where we must act, whether we would 
or no, and where we can never rise in our calculations to 
the level of certainty, there we may confidently trust to 
probabilities ; where on the other hand we are not obliged 
to act at all, or at any rate not obliged to do anything 
extreme and irrevocable, the proper course is not to regard 
our personal convictions, which rest on mere probabilities, 
as sufficient warrant for carrying out our belief in action. 

281. Where we have matters-of-fact given us, with whose 
inner coherence we are in a measure acquainted, and would 
estimate more accurately the probabilities based on such 
coherence, we trench on a field which spreads beyond the 
scope of the general precepts of logic, and in it we must 
rely only upon our actual knowledge of the particular case. 
In regard to future events, however, — and I shall consider 
no others in what follows — we often find ourselves differently 
placed. Of a number of mutually exclusive alternatives we 



HO DE TERMIN A TION OF INDIVIDUAL FA CTS. [Book II. 

may know that one or the other must happen; but not know 
of any ground for preferring one to the rest ; nevertheless 
practical needs may force us to make choice of one, and to 
base our actions on the supposition that it will happen. 
Under such circumstances we can only regard all equally 
possible cases as equally probable in reality. There is no 
other rule by which we can be guided in our judgments. 
Now we disclaim all knowledge of the circumstances which 
condition the real issue, so that when we talk of equally 
possible cases we can only mean those particular cases 
which are co-ordinated as equivalent species in the compass 
of an universal case ; that is to say, if we enumerate the 
special forms, which the genus can assume, we get a dis- 
junctive judgment of the form : if the condition B* is 
fulfilled, one of the kinds f l , f 2 , f 3 . . of the universal con- 
sequent JF* will occur to the exclusion of the rest. Which 
of all these different consequents will in fact occur, depends 
in all cases on the special form d 1 , or <£ 2 , or b 3 . . . , in which 
that universal condition is fulfilled. If we knew this par- 
ticular form of B, say P, we should be able to deduce for 
certain the corresponding value f 3 of the consequent, as 
suming at least that we had discovered the law by which B 
and F are connected together. For our present purpose, 
however, we suppose that we are ignorant of the special 
shape which B will assume if it does really occur ; it follows 
of course that, if B be realised, some one or other of the 
consequents f 1 , f 2 , f 3 . . . must follow ; but from our point 
of view they all remain equally possible, inasmuch as the 
only condition, so far as we know, of their being any of them 
realised, is the validity of B in general, and that holds equally 
good for all and favours none in particular. Let us assume 
for the present that the universal condition B can, if it 
assumes all the variations compatible with its nature, 
produce n, say six different consequents f l . . . / 6 , then 

* [' Bedingung ' and ' Folge,' the initial letters of the English words 
not being convenient symbols.] 



Chap. IX.] 'CO-ORDINATED CASES.' Ill 

the general condition B must be realised in n, i.e. six 
different ways, for each of the equally possible and mutually 
exclusive consequents to be able to realise itself. Thus we 
see that, assuming what is equally possible to be equally 
real, the chance that a particular case will occur admits of 
being mathematically determined ; for each of these f has 
an equal share in the prospect of being realised in a par- 
ticular case, with the others which are equally entitled with 
it to be real ; but the sum of all these probabilities must be 
a constant quantity independent of their number, for it 
must denote the certainty that some one or another of the 
particular consequents^ however many they be, — that is to 
say that ^generally — must occur in each individual case so 
soon as the general condition B is realised in any one 
of its forms. This certainty is equally absolute for every 
B and every F, and only in relation to it do the respective 
chances of the several cases admit of being quantitatively 
determined ; consequently there is no reason for, or ad- 
vantage in assuming the constant in question to have any 
other value than unity ; the chance of any one particular 

case of the n co-ordinated cases / thus becomes = — s and 

J n 



the sum of the n chances = — — = i, or in other words i 

n 

is the exponent of certainty. Thus far I häve used the 
expression 'co-ordinated cases' without explaining it; I do 
so now in order to prevent misunderstanding : a co-ordi- 
nated case is a case which answers to one and only one of 
the mutually exclusive values & l , P, . . of the condition B, 
and these rival values may occur in reality; it does not 
answer to a more general form B 1 of this condition, which 
can never exist in reality, because it embraces several of the 
particular values &\ d 2 . . ; it follows that each of these fs 
is also an elementary and particular form of the consequent, 
without in turn itself comprising other species which can 



1 1 2 DE TERMIN A TION OF INDIVIDUAL FA CTS. [Book lh 



exist apart by themselves, and of which it is merely the 
general expression. For example we may if we choose 
give the disjunctive judgment the form : if B holds good, 
then either f l or F m holds good, by F m being understood 
all the m or n — i consequents f, which are not f 1 ; in such 
a statement f 1 and F m are not co-ordinate terms; the 

chances of f 1 indeed remain - 3 but the chance of F m is 

n 

the sum of the chances of all the elementary cases which 

n — i 

in thought we unite under this formula, and so it = ^ * 

Now it often happens that we are led to institute an 
enquiry by the interest which attaches for us to some 
property which the different cases comprised under F m 
have in common, and for that reason we separate them 
from the rest and denote them by a common name as one 
case, to which we oppose the rest. If we would then 
formulate the probability of this collective case F m , we may 
say it is equal to the proportion, which the number of 
elementary cases combined in it bears to the aggregate sum 
of all possible cases; or we may state it more accurately, 
taking account of the connexion of the whole matter, thus : 
the probability of F m is equal to the ratio, in which the 
number of the variations of B, which may issue in a case 
of the kind F m , stands to the entire sum of all possible 
variations of B ; in a simpler and more general form still : 
the probability of F m is equal to the ratio which the number 
of chances favourable to it bears to the sum of all thinkable 
m 

chances, = — • This fraction is what we understand by 
n J 

the mathematical probability of a future event, and is not 

at the bottom essentially different from, but only more 

accurately determined than the probability of common 

parlance. For usually we say vaguely a thing is probable 

without specifying the degree of probability which attaches 



Chap. IX.] STATEMENT OF CHANCES SUBJECTIVE. 113 



to it ; of two events that one is pronounced absolutely- 
probable, whose mathematical probability is the greatest or 
at any rate usually if wrongly regarded as greatest, the other 
event only appearing improbable in comparison with it. 
In treating of chances mathematically we do not ordinarily 
talk of a thing being improbable, but if we did we could 
only mean that which is relatively less probable. 

282. From small beginnings, which seemed at first to be 
useless except to satisfy scientific curiosity, the calculus of 
chances has developed in the hands of the greatest mathe- 
maticians into an extensive body of doctrine, bearing fruit 
in the most diverse fields of scientific research, besides 
throwing light on many practical questions, the grand logical 
achievement in fact, which the modern spirit of discovery 
has to set over against the wonderful but fruitless theories 
of antiquity. In this form it has outgrown the limits of this 
treatise, and though every detail of it would always be more 
entitled to a place in a system of logic than those useless 
syllogistic subtleties, which in deference to our extravagant 
love of classical literature we have to be always repeating, 
still I am forced to confine myself to the enumeration of the 
simple logical thoughts, which are merely preliminary to 
calculations into which we cannot at present go any deeper. 
But in doing so I am conscious that a gap is left, and must 
point out that this gap needs to be filled, though I do not 
attempt to fill it myself. 

1. In the first place we must make it clear to ourselves 
what we mean by the probability, which we have just learned 
to measure in the simplest cases mathematically. It does 
not imply any positive assertion on our part touching the 
real future occurrence of the event, to which we attribute it; 
it does not express any objective property or nature belong- 
ing to the event, but denotes throughout what is purely 
subjective, viz. the degree of confidence, which we may 
reasonably accord to the future occurrence of a particular 
case, when all that we have given us to go upon in forming 

Logic, Vol. II. I 



1 1 4 DE TER MIN A TION OF INDIVIDUAL FA CTS. [Book IL 



our judgment is the number of cases possible under the par- 
ticular given conditions and not any actual ground carrying 
with it the necessity of one of them to the exclusion of the 
rest. Let, in accordance with § 281, the probability that a 
particular side of the die will face upwards after the throw 
= Y 6 the probability that one of the five other sides will fall 
upwards = 5 / 6 ; then all that these two numbers signify is 
this, that before the throw the trust we may reasonably 
repose in the occurrence of the first case must stand to our 
trust in that of the second in the ratio 1:5; they contain no 
positive prediction that the one or the other will occur, or 
that on repeating the throw the one will occur more fre- 
quently than the other. We postpone the question, how far 
such an inference from the calculated probability to the real 
event is permissible. 

2. If two mutually independent variable conditions B 
and B 1 may lead to n and n 1 different cases respectively, 
the chance that a particular case in the one series will 
coincide with a particular case in the other is equal to the 
product of the chances, which each of the two has in its 

_ m m x 

own series, 1. e. to the product of - • T ? where and nr 

respectively signify the number of favourable chances be- 
longing to each in virtue of the constitution of its condition 
B or B 1 . If two dice are thrown, the side which the one 
shows uppermost has nothing to do with the side which the 
other shows uppermost ; but each die has 6 sides, each of 
which may fall uppermost, and each of these may with 
equal possibility coincide with any one of the six sides of 
the other; there are thus 36 possible cases and the pro- 
bability of each single one of them is V 36 = 1 / 6 . % . If 
however we look upon it as making no difference, which of 
the two similar dice shows the one and which the other of 
two different numbers of points, the probability of any two 
in particular concurring = 2 . Y 36 = Y ]8 ; for if we throw but 
one die or the die B there is of course but a single chance 



Chap. IX.] EXAMPLES OF THE CALCULUS. 115 



of any particular side falling uppermost, but if we throw two 
dice, that is in case of the combination B + B 1 , there are 
always two chances in favour of any two differently marked 
sides falling uppermost together. On the other hand the 
probability that two similarly marked faces will fall upper- 
most together must still remain = Y 36 , for there is only 
one combination which can produce a particular doublet. 
Lastly, if our object be to throw both dice together and 
get a particular number of points between the two, the 
sum 7 has most probability = 1 / Q — 6 / 36 , for it has 6 favour- 
able chances in the combinations 6 + 1, 5 + 2, 3 + 4, each of 
which occurs twice; the smallest probability, viz. 1 j m attaches 
to the sums 2 and 1 2, each of which can only be produced 
in one way. 

Again, suppose we put in an urn i? 1 7 black and 3 white 
balls, in a second urn B 1 6 black and 4 white balls, and 
then ask what chance there is of drawing two white balls, 
one from each urn ; it is evident that in this case as in the 
last what the one hand grasps is quite independent of what 
the other hand has grasped ; but the probability of drawing 
a white ball out of the first urn is m = 3 favourable chances 
out of 20, the probability of drawing a white from the 
second urn is m l = 4 out of 10. Now there are 10 balls 
in B\ and we may draw a white ball from B with any one 
of them ; also there are four whites among these ten ; con- 
sequently the chance of one of these four being drawn along 
with whatever we draw from the other urn would be V 10 ; but 
as the chance of our drawing a white ball from that other 
urn was only 3 / 20 , the chance of our drawing two whites 

711 111} 

together one from each urn will = : y 20 . 4 / l0 = 3 / 5Q . 

We should get another result if we gathered all the balls 
into one vessel and drew twice out of it, taking care how- 
ever to restore the ball first drawn before we drew a second. 
The result of the second draw would then as in the above 
case be independent of that of the first ; for each draw the 

1 2 



1 1 6 DETERMINATION OF INDIVIDUAL FACTS. [Book II„ 

probability of a white ball being drawn would = 7 / 20 , so that 
the probability of two whites being drawn in succession 
would — 7 / BO . r / Z0 = 49 900 > tnat * s t0 sa y would be less than 
in the first case. The difference of the two results may 
seem strangely great, as without calculating it one would 
hardly suppose there was any essential difference between 
the two modes of proceeding ; there is however, inasmuch 
as it is harder or easier to draw one of the white balls just 
according as there are more or fewer black balls mixed with 
them. The chance, 7 / 30 , of drawing a white out of the whole 
collection of balls amounts no doubt to 14 / 9 of the chance, 
viz. 3 / 20 that there is of drawing a white out of the urn which 
contains 20 balls ; for the same reason however it amounts 
to only 7 / 12 of the other chance 4 / 10 , which is the chance of 
drawing another white from the other urn, which contains 
10 in all. Consequently the chance of drawing two whites 
by the second method is only 14 / 9 . 7 / 12 or 49 / f)4 of the chance 
of obtaining the same result by the first method ; we have 
in fact 49 / 5 4 - 6 /ioo — 49 / 9 oo- ^ * s better to be quite clear on 
this point, so I will take a still simpler example. Let us 
assume that the urn B contains but one white and no black 
balls, while B 1 contains one white and one black ; then if 
we draw from B we are certain of one white ball, whose 
probability therefore — 1 ; and we may draw either a white 
or a black from 2? 1 , with either of which it may concur; 
thus the chance of either of these cases, one of which 
consists in two whites following one another is 1 / 2 — 1 . Y 2 . 
Such is the result got by the first method, that of dividing 
the balls in separate urns. By the second method how- 
ever, which consists in putting them all in one urn, we are 
certain of nothing ; for the first as for the second draw the 
chance of a white is the same = % , and that of two whites 
in succession = %, that is to say smaller than it is upon 
the first method. 

3. Suppose the variations of a condition B produce a 
series of cases of the kindyj but the actual occurrence of 



Chap. IX.] THE TRANSMISSION OF EVIDENCE. 117 



one of these cases modifies the condition B 1 which leads to 
consequences of the kind f 1 ; the chance that a particular 
case of the series /will coincide with a particular case of the 
series f l is equal to the product of the independent chance 
of f into that of f 1 as modified by the occurrence of the 
former. We get such a case by slightly modifying the last 
example. If we put back the first ball drawn into the urn, 
which contains 30 balls, we leave the second draw inde- 
pendent of the first ; but if we do not restore it, the urn will 
only contain 6 white out of 29 balls ; the chance of still 
drawing a white becomes 6 / 29 and that of alighting on two in 
succession = 7 / 30 . 6 / 29 , and is only about o-88 of the chance 
which there was of drawing two successive whites, when the 
ball first drawn was put back into the urn. This was to be 
expected, as the number of white balls is now proportionately 
less than that of the black, among which they must be 
sought. Under this head fall many of the problems to 
which the calculus of probability may be applied, and great 
care must be taken to discriminate them from the former 
class. We very often have to do with events, whose chance 
of recurring in the future depends on the number of cases, 
in which on previous occasions either they themselves or 
others standing in a definite relation to them, have been 
realised ; and it is not always easy by analysing this inter- 
dependence to ascertain the influence, which the occurrence 
of one case exercises in conditioning the probability of the 
one to be next expected. 

I have no space to illustrate this by examples, but shall 
give an instance of a different sort. An eye-witness imparts 
something he has seen to someone, who in turn imparts the 
information to a third person. Now we know from experi- 
ence that the further news travels in this manner the more 
distorted it becomes, and accordingly it has been proposed 
to ascertain what degree of trust may be reasonably reposed 
in a statement in proportion to the number of people con- 
cerned in its transmission to us. I do not believe that any 



1 1 8 DE TERMINATION OF INDIVID UAL FA CTS. [Book II. 

amount of calculation will really help us to answer the ques- 
tion. To begin with, it is not quite plain what we are driving 
at. An allegation is either right or wrong ; but if wrong it 
deviates more or less from the truth ; and we might assign 
to it a greater or less degree of credibility according as it 
deviates more or less, supposing it to be possible to measure 
against one another the different amounts of these deviations. 
But this we shall seldom be able to do ; each term of a 
judgment, expressing an original observation, can be taken 
apart and falsified in a way peculiar to itself and when falsi- 
fied can be variously combined with other terms ; the 
aggregate of errors thus arising cannot be regarded as con- 
stituting a series of terms, which we can compare together, 
and we should thus have no available standard by which to 
estimate the objective credibility of the statement as handed 
down. But after all this is not what we really want ; we 
want to ascertain the particular degree of trust which may 
be based on our knowledge of a single condition, which we 
have stated, viz. of the number of times a bit of intelligence has 
been handed on from person to person before it reaches us. 
But here the objection at once occurs that this condition of 
transmission does not in itself contain anything that could 
at all justify us in predicting a gradual falsification of the 
statement transmitted. When, as in the above example, we 
have drawn a white ball and removed it from the vessel, 
which has in it 30 balls, 7 of them white, we know that the 
conditions of a fresh draw are changed and we know exactly 
by how much ; on the other hand if we restore the ball we 
are equally certain that the conditions are the same as before, 
that the second draw is a res Integra so to speak and its 
chances the same as those of the first. It is to the latter 
not to the former case that the problem now before us cor- 
responds; the mere fact of transmission, taken in itself, 
cannot cause me to transmit something else than I have 
heard ; so far as the mere transmission goes there would be 
not a mere probability but an actual certainty that the last 



Chap. IX.] THE TRANSMISSION OF EVIDENCE. 119 



hearer will accurately receive the original statement. Thus 
the falsification of a statement depends not on the number 
of times it has been passed on but on the size and sort of 
errors made in it each time it has been passed on ; conse- 
quently our knowledge of the number of times it has passed 
from mouth to mouth will only help us to estimate its trust- 
worthiness, if the size of the various errors be either constant 
or a regular function of that number. There is not the least 
ground for assuming any such thing; we see it to be quite 
the reverse if we really reflect on the very various cases 
which may occur. The eye-witness A may or may not have 
wished to communicate aright what he has rightly observed ; 
his hearer B has or has not understood him aright, or he 
may have understood him and yet desire to hand it on him- 
self in a distorted form ; a third person C, who intended to 
distort afresh what he already misunderstood, may chance 
to hit upon the actual truth in what he communicates. If 
we consider all these possible conditions we see clearly that 
the trustworthiness of a communication in no way depends 
in any regular manner merely on the number of times it has 
passed from mouth to mouth. We disregard these con- 
ditions because we are ignorant of them ; but if we had the 
power of knowing them all the question would answer itself 
and we should not need to calculate it at all. All we can 
do in the matter therefore is to make utterly arbitrary as- 
sumptions in regard to all these conditions, which would be 
tantamount to bringing the whole thing down to the level of 
arithmetical examples, which had no bearing on or applica- 
tion to real events. Such would be the following sort of 
calculation : say that we hear and in turn report anything 
with such accuracy as to deduct one tenth of its credibility ; 
then after the 20th repetition of the statement its credibility 
would only be o-9 20 = 0-1216, only a little more than % of 
what it was originally. Here all is arbitrary assumption; 
it is arbitrary to assume that the credibility diminishes in 
geometrical progression, instead of arithmetical; the latter 



l 2 o Z>£ TERM INA TION OF INDIVID UAL FA CTS. [Book II. 

is quite as conceivable. No less arbitrary is it to suppose 
at all, that the exponent or difference of term from term 
must be equal ; the result too which we thus reach has no 
meaning ; it might perhaps be true of frivolous street-gossip, 
but as regards serious historical traditions it is a gross 
exaggeration of the rate at which their untrustworthiness 
increases. 

4. Given certain facts we have to conjecture their true 
causes; we must calculate the probability with which the 
given effects would follow from the various possible causes, 
and select that cause as the true one on presupposition of 
which the facts would most naturally follow. I draw four 
times in succession from a bag, and draw from it 3 white 
and 1 black ball, restoring the ball each time, and the ques- 
tion is asked what number of balls of each sort must the 
bag most probably have contained in order to give this 
result. In order to answer the question we must know the 
whole number of balls in the bag, in order that we may be 
able to state the number of conceivable combinations 
capable of causing the given result. Suppose there were 
4 altogether. Now to account for the result at all there 
must be at least 1 white and 1 black ; how many more of 
each kind remains indefinite ; there are 3 possible combi- 
nations, which we can assume : 3 whites + 1 black, 2 whites 
+ 2 black, 1 white -f- 3 black. For these 3 combinations 
the chances of drawing a white are respectively 3 / 4 , 2 / 4 , V 4 , 
of drawing a black Y 4 , 2 / 4 , 3 / 4 ; the joint chances however of 
drawing in 4 successive draws 3 whites and 1 black become 
on these various assumptions 27 / 256 , l % 56 , % 5G j consequently 
the first assumption, that 3 whites and 1 black ball were in 
the bag, is the most likely; at the same time the fractions 
got give the particular probability which each of the other 
two possesses. A very simple consideration confirms this 
solution. Had the bag had in it but a single white, accord- 
ing to the third hypothesis, we must in 4 draws have 
grasped it 3 times, while we onlv once grasped a black ball 



Chap. IX.] 2"HE CHANCES OF RECURRENCE. 1 2 1 



out of three which offered themselves, — a supposition ob- 
viously less probable than four draws in which each ball has 
its turn. It may be noticed that this calculation of course 
presupposes that the different causes, which we can assume 
in order to account for the given facts, in themselves possess 
equal probability; this was the case here so far as any dis- 
tribution of the two colours among the 4 balls was in itself 
quite as possible as any other ; where the probabilities of 
the causes are not equal, due account must be taken of the 
same in our calculations. 

5. When we see the same result repeat itself under the 
same general condition B we are led to expect it to occur 
again if B recurs. The chance of its really doing so admits 
of being calculated. A bag has in it two balls and it is 
found that so often as we draw we always get balls of one 
colour, say white, so that the colour of the other ball re- 
mains unknown to us. Hence we expect to get a white 
upon drawing a third time, supposing ourselves to have 
drawn twice. How shall we measure the probability of our 
expectation ? One ball must be white, so that there are 
only two possibilities, either the other is black or both are 
white. Now two whites have already been drawn in two 
draws, and the probability that this which has happened 
would happen becomes on the first assumption V 4 , on the 
second = 1 ; consequently the odds in favour of the rival 
assumptions stand to each other as 1 : 4 and as their sum 
must be = 1, the first must be put =V 6 , the other = 4 / 5 . 
In case we make a third draw the odds in favour of the 
white are V 2 on tne nrst assumption and 1 on the second ; 
the sum of the favourable chances presented by both as- 
sumptions taken together is thus % . % + % . 1 = 9 / 10 . In this 
case the actual event had occurred and we only knew and 
calculated the a priori chance which it had of occurring 
under two rival presuppositions as to its conditions ; but 
even where we have not this knowledge, we may draw an 
inference as to the chance that an event will recur from the 



12 2 DE TERMINA TION OF INDIVIDUAL FÄ CTS. [Book II. 



number of times we have actually observed it occur. Sup- 
pose we are quite ignorant of its conditions and grounds and 
only know that an event E has once occurred under certain 
conditions, say at some critical moment of time t y it may at 
first sight seem as if the chance of its occurring a second 
time under the same conditions was exactly as great as its 
chance of not occurring at all. But this is a miscalculation • 
were it true the observed fact of its having once occurred 
would be ignored in our calculation, and as the same re- 
flexion might be fairly made after the event had occurred 
for the ;;/ ttl time, we might find ourselves in the absurd 
position of maintaining that the fact of an event having 
occurred, even an infinite number of times did not make its 
occurrence next time any more likely than it would have 
been, if it had never yet occurred at all. This however 
would be evidently paradoxical : for every fresh repetition 
of an event is a fresh and additional testimony to the con- 
tinuance of the unknown causes on which it depends, and 
so strengthens the probability of its occurring again. Our 
conclusion therefore in regard to the first case must be this ; 
that E will not occur is in itself just as likely as that it will : 
but for the existence of causes which bring about E we have 
the testimony of this one observed case of its occurrence ; 
for the existence of causes which prevent E we have no- 
thing but the bare possibility. We consequently have two 
reasons for expecting E to recur, where we have only one for 
expecting it not to, as the two chances stand to one another 
in the ratio 2:1, while their sum must = 1, the chance that 
E will recur = 2 / 3 . In general therefore, if an event E or a 
particular cyclical course of like events E has been observed 
m times without any exception, the probability that E will 

happen again in the same way is = > m tn ^ s fraction 

the denominator represents the sum of conceivable cases, 
since after m real cases have occurred there are always two 
additional cases, which we can think of as occurring, viz. the 



Chap. IX.] 



MATHEMATICAL EXPECTATION. 



123 



repetition or non-repetition of E ' ; the numerator as usual 
denotes the number of favourable chances. I think this 
simple deduction of the formula will satisfy the reader ; it is 
to me as convincing as the more obscure analysis, by which 
it is usually obtained. One sees that as m increases the 
fraction approaches nearer and nearer to unity, and so it 
becomes more and more nearly certain that E will recur. 
The example usually adduced is that as the alternation of 
day with night has been now historically attested for 5000 
years, the probability of the same alternation recurring to-day 
= 1,826,214:1,826,215; that is one may bet 1,826,214 to 
1 on its occurring again. Now if it is true of calculations of 
probability in general that they do not express what will 
actually occur in the future, but only the degree of subjective 
confidence, which we repose in their occurrence ; it is in a 
certain sense doubly true of these cases, as we clearly feel 
when m is a very small number. For then the assumption 
from which we start is that the number m of cases in which 
E has been observed to occur testifies with a certainty pro- 
portional to the magnitude of m to the continuance on the 
next occasion of the causes favourable to E : and this 
assumption is itself but a probability, the strength of which 
is somewhat arbitrary, and of which we only know that it 
increases with the increase of m. Properly therefore the 
formula would not directly measure the probability that 
E will recur, but the probability of this probability, which 
comes to this, that not only the value to be assigned to 
the probability, but also our confidence in this value 
approaches nearer and nearer to certainty as m indefinitely 
increases. 

6. A future event may be fraught with good or evil for us, 
and it is usually a sense of these consequences to ourselves 
which impels us to gauge the strength of our confidence in 
its happening. We shall shape our motives and actions 
according to its strength, and these will therefore depend 
doubly on the likelihood of the event E and on the com- 



12 4 DE TER MI NA TION OF 1ND1 VIDUA L FA CTS. [Book II. 

parative amount of the advantage we hope to derive from 
it. If we multiply the probability of E into the amount of 
attendant advantage we get what we term mathematical ex- 
pectation^ which thus admits of being precisely determined. 
Let a game be so arranged that the player gets two thalers 
if on the first toss he throws heads and five thalers if 
he throws first tails then heads. The probability of the 
former case is — V 2 , its expectation = V 2 . 2 ; the probability 
of the second case is = 1 / 4 , its expectation = 2 / 4 . 5 ; lastly 
the aggregate expectation of winnings when play begins can 
only be the sum 9 / 4 of these two expectations ; for though 
the two lucky cases exclude one another according to the 
arrangement, yet the expectation of winning must clearly be 
greater when both the two prizes are offered than when only 
one, and the expectation of the one must be exactly left 
over, if the expectation of the other is reduced to nothing 
by the gradual diminution of the prize assigned to it. The 
same reasoning would apply, if it were agreed that the player 
should receive two thalers if he threw heads the first time, 
and then another five in case tails followed. The two 
winning cases are then compatible with each other, but here 
too all that can be won is either two or seven thalers, and 
the chance of winning either is V 4 . In this case then as in 
the former 9 / 4 of a thaler represents the aggregate expecta- 
tion of the player and the utmost he can reasonably stake 
upon it. Suppose again that of different events E y E\ E'\ 
which we may expect, some are fraught with evil, others with 
good consequences to us ; in that case it is easy to see that 
the aggregate expectation which we may entertain, if by our 
own actions we are willing to risk their happening, must 
equal the difference between the sum of the mathematical 
expectations of the favourable events and the sum of the 
expectations of the unfavourable ones. If this difference be 
a negative quantity it expresses the magnitude of the risk we 
run or, more correctly speaking, the magnitude of the anxiety 
we should feel. This principle is wide and important in its 



Chap. IX.] RELATIVE MATH. EXPECTATION. 125 

applications ; by means of it we are not only able to deter- 
mine what bets and games of chance are fair and equitable 
— a sort of calculation we could as well do without as we 
can do without its object — but it also assists us in arranging 
the most serious public and private business, such as the 
management of finances, the undertakings of trade and the 
organisation of all sorts of insurance companies. 

7. This is the place to mention one other idea. Even 
the mathematical expectation of an event does not deter- 
mine its value for us irrespectively of our own condition 
before it occurs ; in judging of its real value for us we must 
take this condition into account. A moment's joy to the 
miserable or a trifling gift to the poor is of greater value 
than a fresh triumph to the fortunate or splendid winnings 
to the rich. No doubt as a matter of fact one that has 
much is wont to desire so much the more, but in this 
respect logic takes the point of view of equity, according 
to which it ought not to be so : in assuming as a self-evident 
principle that the relative value of an advantage bears an 
inverse ratio to the advantages of the position to which it is 
added, it expresses the standard according to which a man 
seems to be justified in desiring to improve his condition, 
when the good things available for this improvement have 
also to satisfy the wants of others. This general law does 
not admit of being mathematically applied, unless all the 
advantages of a situation and all the good things requisite 
in order to better it admit of being mathematically com- 
pared ; it is therefore chiefly of use in regard to the increase 
of a capital, which can be expressed in money. Let V be 
a capital which we have already got, and z the addition 
which it is to receive : then this increase of V may always 
be regarded as the sum of an infinite number of smaller 
increments each of the size d z ; the relative value however 
of each subsequent (n+i) th - augmentation by a d z is in 
inverse proportion to the size of V as already enhanced 
by the preceding increments, that is to F+ndz, and would 



l 2 6 D E TERM INA TION OF INDIVIDUAL FACTS. [Book II. 



thus = Tr ,' — j- . In this formula k is a specific coefficient, 
V + ndz r 

which differs with the different sorts of cumulative advantage, 
but is constant for all z of the same kind and does not 
admit of further determination in the abstract; and as it 
forms a common factor in all values which we can compare 
we omit it in what follows. The relative value of the 
aggregate increase by z is then the integral of this ex- 
pression, in which we must replace ndz by values of z 
ranging from o up to z : it is therefore 

= log(V+z) - log V. 
In accordance with this formula we should find that for a 
capital V — iooo the relative values of the increase when 
z = iooo, = 2000, = 3000, = 4000, are approximately 1, 
1 . 6, 2, 2 . 3 : that is to say, they grow very much more 
slowly than do the increments themselves. For the different 
capitals V = iooo, = 2000, = 3000, = 4000, the relative 
values of the increase, when each is augmented by z = 1000, 
are approximately 0-301, 0-176, 0-125, 0-097. When we 
have thus calculated the relative values of the advantages 
which some event will bring us, we may multiply them by 
the chance there is of their ever being attained which we 

V+z 

will call m : by this means we get m log ( y ) for our 

moral expectation of them, i. e. the mathematical expectation 
of these advantages reduced to their relative values; and 
this is what in all sorts of enterprises determines the amount 
of risk we may prudently incur in view of some prospective 
advantage. We have assumed the factor m to be constant 
however high z is; it may be so, but it may also be a 
function of z or of V + z, in which case of course it is to 
be included under the integral sign and brought into the 
integration. In point of fact there are many sorts of under- 
takings in which while the first success is hard to win, 
subsequent successes become easier and easier, or in which 
the possibility of further success diminishes with the increase 



Chap. IX.] 



FUTILE APPLICATIONS. 



127 



of what has been won. Lastly the formulae do not help us 
to measure all that one may wish to measure. By treating 
z only as the sum of d z, without taking account of the 
time / which it takes to achieve the summation, they neglect 
the distinction between gradual and sudden improvements. 
The real, actual, or physical values of the two may be the 
same, but their psychical effect, or, to put it simply, the 
pleasure they occasion, may be different, and this after all 
enters as a factor into the idea of the comparative value of 
an advantage. Let us assume first that the extent to which 
a particular satisfaction admits of being further enhanced 

= ~, if V represents the degree of satisfaction of the same 

sort to which one has already attained ; and secondly that 
the increase in the satisfaction generated remains propor- 
tional to the size z of the sudden increase in the advantage ; 
z 

then — will measure the pleasure due to the accession of z. 

But it is easy to see that these are not the only conceivable 
assumptions; it might even possibly be found that the 
eventual enjoyment is also a function of tn, i.e. of the 
chance that z will occur : we might perhaps be more deeply 
affected on winning a satisfaction, of which we had almost 
despaired, than on winning one of even greater comparative 
value that was more probable. 

283. The last observation just touched upon problems 
which have not as yet been brought within the range of 
calculation, though there is nothing to prevent their being 
so brought, if an advance in psychological knowledge should 
ever afford us starting-points from which to grapple with 
them. Other problems there are to which it is but an idle 
play of words to try to apply the calculus of chances. For 
although this method of inference does start from our 
ignorance of the special grounds, which condition a par- 
ticular event, still it makes certain presuppositions, which 
we cannot neglect. In the first place it presupposes the 



12 8 DE TERMIN A TION OF INDIVID UAL FA CTS. [Book 1 1. 

truths logical and mathematical, of which we must make 
use in order to be able to calculate at all. The truth of 
special laws, limited in their action to a group of facts, the 
non-existence of which is just as conceivable as is their 
existence, may, as we shall presently see, be proved by 
means of calculation; but from what basis could one 
legitimately start to show the law of identity or the doctrine 
of disjunctive judgments to be more or less probable ? The 
very simplest determination of any probability presupposes 
a disjunction of all possible cases to be given, that each of 
these cases is identical with itself and not the same as any 
other, and that each of them is exclusive of the rest. It 
follows that before we set about to prove an event, or a 
state, or a series of events to be probable, we must have 
presupposed the particular content in question to be part 
and parcel of a world, in which universal laws demarcate 
what is true from what is untrue, what is possible from what 
is impossible, what may easily occur from what may not. 

But the calculus of probabilities is subject to other 
limitations besides these. The object, which its problems 
concern, must be regarded by it as not merely thinkable 
in the abstract; it must also presuppose the presence of 
conditions, which necessitate the realisation of one of the 
disjoined cases to the exclusion of the rest ; to use the 
language of its formulae there must always be a certainty 
= i, which is the sum of all the probabilities of the par- 
ticular cases, which we can think of. This was noticeable 
in our examples throughout. If & die has been thrown or 
if two have been, we can determine the respective chance 
of any of the particular cases which may result ; but unless 
we specify how many dice are to fall and how many times 
in succession, it is quite impossible to determine the scope 
of the disjoined possibilities and the unity by reference to 
which the chances of each are to be severally measured. 
It follows that we can only calculate such events as depend 
on one another within a regularly ordered world ; ultimate 



Chap. IX.] CHANCES DEPEND ON REALITY. 



129 



facts, which contain an independent absolute being of their 
own, we cannot calculate. It would be mere senseless play 
of wit to reason thus : prior to all existence there is the 
same chance of the existence of something as of nothing ; 
but one or the other must take place ; therefore the chance 
of something existing is =Y 2 : but this something would 
necessarily be either one or many; consequently the chance 
of there being many elements is Y 4 , and the chance of 
there being one is V 4 : lastly assuming that there are n 
elements, they may be all alike, or some different, or all 
different; the case of these being all alike would be but 
one of the m cases which would thus arise, and consequently 

its probability is = . Prior to all existence, we must 
4 m 

observe, there can exist no ingenious spirit to institute such 
a calculation of what will be ; could we conceive however 
of such a spirit as existing outside the world and speculating 
as to whether it is likely to come into existence or no, still 
that nothing would involve no condition of any sort, which 
would necessitate a real settlement of the alternative pre- 
sented in thought between being and not being, so that 
the end of the whole matter would still be nothing. But 
suppose the alternative to have been somehow settled in 
favour of being, this being could not possibly be some- 
thing, which were merely thinkable in abstracto) it 
must be capable of existing and can only be some 
determinate being, which excludes all other thinkable 
being. Such determinate being would from the very 
beginning have a certainty of its own = 1, while the 
probability of all other kinds of being would be not 
exactly = o, so much as an idea without any assignable 
meaning. It would be different if we wished to determine 
the probability of these ultimate facts from given data : 
on the assumption that all reality is bound together by 
law, these given data would (not as the ground of their 
reality, but as the source of our knowledge of them) con- 

Logic, Vol. II. K 



130 DE TER MI NA TION OF INDIVID UAL FA CTS. [Book II. 

stitute a condition which would compel us to assume the 
one or other form of those ultimate facts to the exclusion 
of the rest. 

284. There is one other point we shall do well to bear in 
mind. In probability we have after all no more than a 
measure of the confidence we may legitimately place in the 
occurrence of an event, before it has occurred. After it has 
occurred however what was previously its greater or less 
probability does not continue to attach to it as a permanent 
property, from which we may regressively draw some other 
conclusion in regard to the causes of its being realised than 
just this that they have actually come about. We are 
victims of all sorts of illusions on this point. For example 
an event E occurs, which on previous calculation was very 
unlikely to occur as compared with a whole class of cases, 
which for the convenience of thought we gather into one 
and collectively oppose to it as a second rival case non-is ; 
in such a case we are apt to imagine that not only a special 
and peculiar but a higher cause was necessary in order to 
bring about E. To take a common instance : some object 
is of little significance, almost unknown and seldom spoken 
of. We stumble on its name once, after which it meets our 
eye again and again in conversation, in books, in periodi- 
cals ; here is a coincidence, the chance of which, if calcu- 
lated beforehand, was infinitely small and we call it a very 
strange incident. But a moment's reflexion will convince 
us that there is very little strangeness about it ; how in- 
finitely more numerous are the cases in which the incident 
does not turn up than those in which it does ! How many 
names just catch our ear once without ever being repeated 
in such a way that their repetition strikes us as odd ! To 
put what I mean in a perfectly general way, let us suppose 
there to exist some condition B or some group B of different 
but co-operant conditions ; these according to the different 
and in the abstract equally possible variable attitudes, which 
they can assume towards each other, would bring about a 



Chap. IX.] THE ODDS AGAINST A FACT 



number n of different results E\ the probability of each 

individual E is then = ~ and by consequence the same as 

that of every other determinate E, but, if n be infinitely 
great, infinitely small as compared with the probability that 
any one indifferently of all the remaining n— i events will 
happen, which we collectively oppose to it. But the latter, 
the collective probability has another significance than the 
former — the individual, for all the n — i events cannot be 
realised, but only one of them to the exclusion of the rest. 

A famous instance will illustrate the extent to which we 
may be led astray by such false comparison of things, which 
are essentially different. The planetary system, according 
to Laplace, and so far as he knew it, consists of 1 1 planets 
and 1 8 satellites; we are acquainted with the revolutions of 
the sun, of 10 of the planets, of the moons of Jupiter, of 
Saturn's ring and of one of his satellites 3 the rotations of 
these bodies together with their revolutions form a group of 
43 movements in the same direction ; would we assume 
that this uniformity is all a matter of chance, we find on 
calculating it that the probability of such an assumption 
being true is something less than unity divided by four 
billions. I have no doubt that fresh advances in astro- 
nomical science will leave this number substantially correct, 
but what follows ? Simply that the particular cause or group- 
ing of causes adequate to produce this state of things is or 
has been real. It does not follow that this grouping of 
causes itself requires any other cause than just that so-called 
chance, by which we mean no more than that the mutual 
relations of several realities, which we presume to form a 
group, may without contradiction be combined in an infinite 
number of ways. Out of all these four billions of possible 
cases never more than a single one can be realised, and no 
matter which of them it be, we shall feel just the same 
surprise at its happening in particular as we should have at 
any other of the four billions, had it happened. The case 

k 2 



132 DE TER MIN A TION OF INDIVIDUAL FA CTS. [Book II. 

would be utterly different, if all those other dispositions of 
events really formed a single second case, capable of being 
realised as such ; in that case its chance, denoted by four 
billions, would have at least admitted of being directly 
compared with the probability of the other case, i.e. with 
unity, although even then the sort of conclusion would not 
have been justified but only rendered more attractive. No 
doubt a plausible attempt may be made to justify the re- 
duction of the whole number of cases to a single pair of 
alternatives ; nothing, it may be said, but this given disposi- 
tion of all masses and motions could secure the stability of 
the planetary system and the continuance of its movements ; 
no one of the million other arrangements would have served 
to produce this state of equilibrium. We concede all this, 
but might it not be also pleaded in favour of each of the 
other arrangements, that it too had in store for the planetary 
system a particular destiny of its own, that on presupposition 
of it alone among the many millions of possible presupposi- 
tions could that peculiar destiny be fulfilled, so that if the 
uniqueness of the result constitutes a claim to a higher 
origin each of these rival dispositions may with equal right 
prefer such a claim ? It would seem then as if it was not 
the unique result as such, but the unique result which was 
better than its fellows, which finally prevailed. But why 
should superiority in itself constitute improbability ? And 
after all tvould this case which ultimately prevailed be better 
than others? No doubt as things are our blood is ever 
fresh and new, yet at the expense of for ever circulating in 
the same forms ; but is it really and without qualification a 
finer thing that it should circulate as it does than that it 
should not ? The perpetual repetition of these forms may 
no doubt appear very grand to us to-day, to-morrow may it 
not strike us as rather tedious? Would it not be a finer 
thing if the planetary system were not in such stable 
equilibrium, if all its relations were for ever changing, so 
that vegetation and natural beauty, animals and man should 



Chap. IX.] 



% IN THE LONG RUN' 



133 



develope in ever new and interesting forms and history be 
really the history of a progress, of a manifest advance, 
instead of the chronicle of a cycle of ever-recurring events ? 
And to conclude, inasmuch as the heavens are infinite, 
may not all the millions of differently ordered systems be 
actually realised therein ? With us the system of equili- 
brium, at unknown distances the rest? And then surely 
our own system would only possess such reality as its 
probability entitles us to claim for it, it would be but one 
among millions. 

285. Thus far we viewed the calculation of chances as a 
mere means of ascertaining accurately the confidence we 
may repose in the occurrence of future events. There now 
arises a natural desire to know how far these previous calcu- 
lations are as a rule confirmed by the actual course of 
events. The answer usually made is that the more numer- 
ous the cases, which make an event F possible, the more 
closely does the number of times it actually occurs tend to 
coincide with the number calculated. We can only get an 
answer that is at all trustworthy by means of experiments of 
the simplest kind, in which care is taken to restore after 
each nifo experiment the group B of conditions, upon which 
each particular case ^depends, in such a way as to leave 
the composition of B exactly the same after as before that 
experiment, differenced only by the variations, whose in- 
fluence on the net result of the series of experiments it is 
the very object of our enquiry to ascertain ; taking care 
at the same time to prevent the entrance of any alien cause 
not implied in the idea of such variation, whether that cause 
consists in external circumstances or in a change of the 
object of the experiment or in unfair intervention, on the 
part of the person experimenting. These conditions are 
fulfilled in experiments made with dice. We calculate before- 
hand that, if two dice are thrown once, the chance of getting 
a particular combination of points, e. g. 5 . 6 = V 18 = 0-056, 
which for a thousand throws would be 56 ; if now 



134 DETERMINATION OF INDIVIDUAL FACTS. [Book II. 

we make trial of these iooo throws one after another, and 
find, as in fact has been found to be the case, that the 
specified combination occurs 50 times, we see that this 
number already approximates pretty clearly to the number 
calculated ; yet more when in 10,000 throws it rises to 570. 
Each single throw in such a case depends, if we leave out 
of account the uniform or changing resistance of the air, on 
the following conditions : on the velocity with which and 
the angle at which the die impinges on the receiving plate, 
upon the position of its sides and corners at the moment of 
impact, upon its own elasticity and on that of the plate. 
We may regard the last of these conditions as constant, for 
as we should expect from a calculation of its probability, 
the die will extremely seldom touch the same point of the 
plate, so that the elasticity of the point of impact will not 
change to any appreciable extent, if it was the same to begin 
with for all points of the plate. If, however, we would still 
regard it as variable, it may be included, just as well as the 
slight and gradual changes in the shape and elasticity of the 
die, among the variations of the conditions, the effect of 
which is being investigated ; for since the two changes do 
not depend on each other, but may co-operate, they do not 
when taken together favour one particular throw more than 
another, but favour now one now another indifferently. 
The first-named condition, the velocity and direction of the 
die, depends of course on the movement of the hand that 
casts it ; but were one even disposed to favour a particular 
throw by this means, one could hardly do so effectually ; 
for after we have obtained a particular throw, we neither 
retain a clear recollection of the group of muscular feelings, 
which accompanied it, nor are we able to reproduce the 
exact movements, on which those feelings depended, so as 
to exactly copy the throw ; and the least deviation would 
have the effect of favouring some other combination of 
points than what we wished to throw. These very changes 
therefore of our movements are among the legitimate 



Chap. IX.] RESULTS NEED NOT AGREE. 135 



variations of the conditions of the result we are investi- 
gating. 

The same advantages are presented by a rotating drum, 
into which m white and p black balls are introduced and into 
which each time we have drawn we re-introduce the ball 
drawn before drawing again. If we then turn the drum we 
do not of course restore exactly the same position to the 
balls, which they had before we drew, but still we only 
produce one of the variations of this position, with the in- 
fluence of which we wish to become acquainted. If we 
distinguish the whole condition B, upon which the event F 
in each separate case depends, into a constant and a variable 
element, we may say that in the first case the shape of the 
die forms the constant element as does the number of black 
and white balls in the second, while the variable element 
consists in the first case of the velocity and direction of the 
die, in the second case of the relative positions of the balls 
and the direction given to the hand in drawing. If we 
actually make the second experiment the result obtained is 
similar ; the greater the number of draws the more nearly 
does the ratio between the numbers of the white and black 
balls drawn approach to the ratio between the numbers m 
and in which they were present in the drum. 

286. Theoretical considerations have been based on the 
results of these experiments, which I cannot persuade my- 
self are correct. A vicious circle is involved in all attempts 
to show that the results mentioned occur always with an 
intelligible necessity. In the first place one cannot argue 
from m series of experiments, in which we have really 
obtained it, to its being obtained in every (^ + i) th series, 
so long as the unknown variations of the conditions, which 
have there produced the said result and would produce it 
here, are individually subject to no rule whatever. For the 
idea that they will at least on the ivhole continue to com- 
pensate themselves in the same way here as there — and it is 
only on this condition that the attempted universalisation 



136 DE TERMINA TION OF INDIVIDUAL FA CTS. [Book II. 

of the observed results would be permissible — this idea, I 
say, has no objective validity, nor is it to be deduced from 
anything we already know to be real ; it is indeed simply a 
way of expressing our subjective and almost tautological 
maxim, that that is most likely to occur in reality, which 
previous calculation shows to be most likely. Provided, 
that is to say, no uniform cause gives the preference to one 
of the possible cases of the kind F over another, we must 
ascribe the same chance of being realised to all cases, which 
in their idea are co-ordinate or equally possible ; in which 
case all that is meant by saying that some particular fact or 
event is to be expected as most probable is just this, that in 
a great number of experiments the actual number of times a 
case F occurs is equal to the calculated number. If this 
expectation is verified in m series of experiments, that has 
actually occurred m times, which before the occurrence was 
the most probable : but it does not therefore become a 
demonstrable necessity that it should again occur in every 
(w+i) th series of experiments; that indeed remains the 
most likely thing we can expect, when we are brought face 
to face with this new series of experiments, but it may 
always turn out to have been a wrong expectation. 

In the second place no single series of experiments can 
really comprise an infinite number of experiments ; it must 
always stop short at some finite number, however large 
it be. Thus it can never be a real fact of observation, 
that the number of actual realisations of F approximates 
without limit to the calculated number as n increases : it 
is always an inference from the facts. Now assuming 
that in n experiments we reach a point at which the two 
numbers coincide or that they have so nearly coincided 
that their difference need not be considered, it would 
be a very arbitrary procedure to break off the series just 
at this point. It is obvious that the law of such equality or 
approximation comes true if we continue the series till 
it comes true and no longer. But what if we prolong the 



Chap. IX.] INFINITE SERIES NOT REAL. 



137 



series? Possibly the theoretical and the real results will 
converge still more ; possibly again each additional round 
of n experiments will have the same or nearly the same 
result as the first had, and the difference d will not be 
sensibly diminished by prolonging the series : and to these 
possibilities we may add any other less regular succession 
as also possible. Only these different suppositions have 
not the same amount of probability ; so long as we con- 
sistently avoid presupposing any constant cause, which 
in a series of experiments to be made might give one 
case F a preponderance over others, our most probable 
assumption is merely, that as n goes on increasing the 
number of observed realisations of F will continually ap- 
proximate to the number of them calculated beforehand. 
If in a large though limited number n of experiments 
this expectation is not verified, a constant or uniform 
condition may be chargeable with the result, though it 
may also be due to the combination on no principle of 
variable conditions. As often however as our expectations 
are verified by what really occurs, we are presented with 
a fact) at which we cannot feel any surprise just because 
it was beforehand not improbable, but still a fact of which 
we can just as little prove that it was necessary as we 
can prove of the verification of any mathematical chance 
that it was necessary. In the experiment with the drum 
and balls an uniform proportion gradually revealed itself 
between the numbers of the differently coloured balls 
drawn; I cannot believe that this uniformity is really 
explicable, if that means anything more than probable. 
The distinction made between the constant and the vari- 
able or accidental causes which jointly produce an effect 
is a very true and significant one, but this is not in my 
opinion the place to appeal to it. It is argued that 
however irregular the successive arrangements of the balls 
may be, there still remains one constant element, namely 
the unchanging proportion of white to black balls ; this 



138 DE TER MI NA TION OF INDIVIDUAL FACTS. [Book It 

in a great number of draws must make itself felt by 
producing some constant effect : for there is no reason 
to suppose that just where the hand alights balls of one 
colour will be found oftener than in proportion to their 
relative number : if such were found to be the case, we 
must, it is argued, break with our supposition and assume 
some constant collateral cause which favours that colour. 
Against this view it may be objected that the constant 
causes spoken of could not make themselves felt by merely 
being there, but only so far as they act. In the experiments 
with dice the shape of the die and the position of its 
centre of gravity were such constant causes and both took 
effect in each single case. In virtue of the former the 
die could only fall on six sides and not on a seventh, 
in virtue of the latter it could not help falling on one of its 
sides instead of coming to rest on one of its angles or 
corners ; on which of its sides it should fall, however, was 
just what these constant causes did not determine. It is 
the same with the experiments with balls in a drum ; two of 
the conditions are constant; firstly the colours are only 
white and black, so that no blue or red ball can be drawn ; 
secondly their numbers m and p are constant, though the 
relative numbers of the few balls which come within reach 
of the hand each time it draws are none the less to be 
classed as variable elements in the condition ; hence it 
follows that this constant condition, the ratio m : does not 
take effect, though it actually exists. I do not therefore 
see the necessity of assuming — what contradicts our pre- 
supposition — a constant collateral cause to account for the 
apparent anomaly of a different proportion of balls being 
drawn to what is in the urn. On the contrary all that is 
needed to produce such a result is that the positions of 
the balls should be changed on no principle, and such 
irregular change is just what we presuppose and try to 
bring about by turning the drum. Such change of their 
positions renders possible every and any combination of 



Chap. IX.] STATISTICAL GENERALISATIONS. 



139 



the balls, makes it possible even that all the balls of one 
colour should be missed and even that this exclusion of 
one colour should be repeated over and over again in 
successive experiments, everything being designedly so ar- 
ranged that each (m+ i) th experiment is entirely independ- 
ent of the m th . All that can be said against this is that 
it is the reverse of probable; all that is probable is that 
the number of times a ball of a particular colour is grasped 
will tally with the number of balls of that colour which 
there are in the urn. But this too is no more than a 
probability ; if it be nearly verified by experience we have 
got a fact, not inexplicable, inasmuch as we see quite well 
how easily the causes which contribute to that result may 
come together, but not explicable in the sense that one 
could demonstrate both that and how they must thus coin- 
cide in the long run of cases, whereas, when the cases are 
few in number, there is no must about it. 

287. In the foregoing examples we were cognisant of 
the nature of the constant causes as well as of the extent 
to which those which were not constant might vary ; hence 
we could in anticipation of experience make assumptions 
as to how often an event they conditioned would occur, 
and find our assumptions verified by experience. We now 
turn to events of which we know neither the constant nor 
the variable causes, but which we observe to occur over 
and over again. What conclusions can we draw from the 
regularity with which they occur? Here we know neither 
how many are the cases, which are barely possible, nor how 
many chances in favour of the event in question there may 
be among them. The only distinction we make is between 
the occurrence or non-occurrence of JS, regarding as cases, 
in which its occurrence is possible, all those which realise the 
particular centres of relation, which being given make E 
intelligible, and comparing with the numbers of these the 
number of cases in which it is realised. The constant and 
variable causes on which blindness depends are hidden 



1 40 DE TER MI NA TION OF INDIVIDUAL FACTS. [Book II. 

from us ; but the number of cases in which this defect 
may owing to those causes occur is equal to the number of 
the population. If we compare with the entire number 
of persons belonging to one generation the total number 
of blind people among them and conceive of this com- 
parison being extended over several generations, we can 
quite see how it would transpire, whether such a constant 
ratio is on the whole to be found between the two numbers, 
as would point to the presence of a group of constant causes 
favourable to blindness in the mass of men, but in the in- 
dividual modified in their effects by variable causes. But 
this is not all ; in most cases it will probably need a con- 
siderable length of time for the variable causes to realise 
themselves by turns in such completeness that they cancel 
each other's influence and allow the constant ones to assert 
themselves. Hence it is usual to try and discover units of 
time, in which the ratio of the actual to the possible cases 
of E becomes the same, due regard being had of course 
to the periodical changes to which the number of the latter 
are liable. Now the year happens to be the particular unit 
of time during which most of the variable conditions, which 
affect men at all generally, run through the cycle of their 
possible values, and so the first question to ask in investi- 
gations referring to human affairs is naturally this : within 
these units of time does the ratio of the actual to the 
conceivable cases of E remain uniform or approach to 
uniformity? The answer to all these questions may just as 
well be negative as affirmative. If an event E occurs at 
all frequently during a certain period of time, there must 
be within that period some constant cause of it, at least 
in the sense that some ratio exists, which to a definite 
extent promotes the combination of variable causes which 
favours E. Then as often as an unit of time recurs and 
the same proportion of real cases to possible ones is repro- 
duced, so often are we warranted in the regressive inference, 
that that constant cause has existed. But it is not at once 



Chap. IX.] THESE RESULTS ARE NOT LAWS. 14 1 

clear how we can argue forwards, that for the next equal 
interval of time the same proportion will hold good as a 
predetermined law. Such an assumption can only be re- 
garded as the safest rule to go by in judging of the future, 
when no data are known pointing to the intervention 
of some change in those unknown conditions. If the 
rule holds good as foreseen we are justified in once 
more making the same regressive inference as before, 
and sure enough the oftener we can make it, the 
oftener that is the rule is confirmed by the facts, so 
much the stronger becomes the probability, that the 
group of conditions, which has remained constant over 
so many units of time, will remain unchanged for the 
future. To more than this probability however we can 
never attain, and so it is very unsafe to characterise the 
results of such observations as laws of what happens, or 
actually to speak, as we sometimes do, of laws of big 
numbers, as if the mere bigness of a number of compared 
cases must of necessity introduce a regularity in the course 
of a certain class of events, which has no independent 
foundation in the nature of those events and their con- 
ditions. A law, as we have seen, is an hypothetical judg- 
ment and enunciates the necessary validity of a consequent 
provided the antecedent be valid. Statistical laws must 
not aspire to satisfy this definition or they certainly lose 
their value ; for they say no more than this : if in the next 
unit of time T all known and unknown conditions be as 
in the last, then will the series of all the consequences, 
consequently also the sum total of E, be the same. Of 
course it will, for if we suppose the past to take place 
over again, it will wear just the same aspect as it wore then. 
Those who talk about statistical laws no doubt do not mean 
to be guilty of tautology ; on the contrary they mean to 
state their antecedent clause categorically, that is, to assert 
that such an identity of all conditions will take place ; 
it is obvious however that such an assertion can never 



142 DE TE RHINA 7 ION OF INDIVIDUAL FA CTS. [Book II. 



be certain, but only probable. Such propositions there- 
fore are not laws, but analogies, which extend a proportion, 
which has held good in n cases, to the (/z-f- i)^, not proving 
but only assuming that between n and n -f i there is no 
change in the conditions on which their validity depends. 

288. Among events, which in their frequent recurrence 
depend at once on constant and on variable conditions, may 
be properly classed our own observations, by which I mean 
the simplest kind of observation, viz. the measurement of a 
quantity given in an act of perception. Here the constant 
cause consists in the true value of this quantity, for this 
under entirely similar conditions would always have the 
same effect on our susceptibility. The variable causes are 
the external circumstances and the changes in our psychical 
state, which modify that effect in different ways on different 
occasions of its repetition. To elicit from the different 
measurements thus obtained, the true value of the thing 
measured, would be impossible if we ascribed to the 
measurements made every conceivable degree of inaccuracy : 
that would mean that we thought we might substitute for 
the values found any others we liked as more correct, and 
that would be stultifying the very idea of a measurement. 
We therefore presuppose that knowledge, aptitude, and 
attention have combined to make the measurements fairly 
trustworthy and only leave a chance of error very small 
compared with the magnitudes themselves which have to 
be measured. Now suppose we wish to determine some 
single unknown quantity A, at first every isolated measure- 
ment we have of it must pass muster as a true determination 
of A ) for even if we had doubts of its accuracy we do not 
know how far or in what direction to rectify it and have 
no grounds to go upon. On the other hand though the 
quantity A can only be one and the same, observation may 
give us different values for it; we have then no absolute 
ground for trusting one value more than another, and as 
we must now suppose all our observations to be more or 



Chap. IX.] PROBABILITY OF ERRORS. 



less erroneous, we are most probably right in fixing its true 
quantity at a value, whose assumption involves the least 
sum of errors in the measured values. The arithmetical 
mean M, i.e. the sum of all the measured values divided 
by the number of measurements, is thus to be regarded as 
the most probable value of A. The difference between 
this mean M and the true value A is the residuary error, 
which, so long as we have no other accessory conditions by 
which to determine A, cannot be got rid of, but only 
reduced by multiplying the number of equally careful 
observations. 

On the other hand if we have repeatedly measured 
different quantities ABC, and are still furnished with other 
conditions, which the values thus obtained must satisfy, it 
may be that the different arithmetical means, which would 
individually give the most probable values of ABC, will not 
collectively satisfy these accessory conditions, and so stand 
in need of rectification. For example, we may have re- 
peatedly measured the three angles of a triangle and found 
that the sum of the mean values thus obtained amount to 
i8o°+ d°\ this d° being incompatible with the nature of a 
triangle will point to an error in the result, which must have 
arisen out of errors in the measurements and can only be 
got rid of by altering the values found. But the reduction 
required may be distributed in very various ways among the 
three measured angles ; and the question arises, what sized 
error may be most probably ascribed to the measurement of 
each angle. This suggests an enquiry based on principles, 
which if not demonstrable a priori are at any rate very 
probable and in harmony with experience, — an enquiry into 
the relative probability of the occurrence of errors in our 
observations generally. In the idea of a careful observation 
as such there is nothing to imply error at all. The chance 
therefore of our having hit on the truth is always greater 
than the chance of our having fallen into any particular 
error. Similarly it is involved in the presuppositions, on 



144 DE TERM INA TION OF INDIVIDUAL FA CTS. [Book II. 



which the eliciting of true values from observations always 
depends, that the chance of large errors is less than that of 
small ones, and the chance of positive errors exactly equal 
to the chance of negative errors of the same size. This 
suggests one way of picturing the problem. Let a straight 
line be chosen as the axis of abscissae, take in it any 
starting-point to correspond to a total absence of error, and 
from this point let there be divided off in opposite directions 
abscissae ±a, +y, in ascending order of magnitude. 
At the point of no error or zero-point draw to the line an 
ordinate of any length, symbolical of the chance of there being 
no error : this will be the longest of all the ordinates, and 
all the others, drawn at the points ±a, ± ß . . will diminish 
in length symmetrically on both sides of it according as the 
errors symbolised by «ßy, the respective chances of which 
they denote, increase in size. But experience at the same 
time teaches us that the chance of errors does not simply 
decrease in the same proportion as their size increases. So 
long as the errors are trifling the chance of them decreases 
less rapidly than their size increases, but the greater they 
are the faster does their decrease in probability outstrip 
their increase in size. Hence the line joining the upper 
extremities of all the ordinates cannot be formed of two 
straight lines, meeting over the zero-point and symmetrically 
approaching the axis of abscissae on both sides, in such a 
way as to form a triangle with a segment of the axis. On 
the contrary the line in question is a curve, the vertex of 
which lies above the zero-point and which branches out 
therefrom in two symmetrical limbs, which are concave 
towards the axis of abscissae. The course of the curve is 
thus easily followed in the neighbourhood of its vertex, it is 
not so easy to follow as it approaches the axis. We may 
regard errors of any size as possible, errors of even infinite 
size ; these too will have their degree of probability, in- 
finitesimal though it be ; in consequence each limb of the 
curve must ultimately become convex toward the axis of 



Chap. IX.] METHOD OF THE LEAST SQUARES. 145 



abscissae and approach it asymptotically. But we need not 
take into account such infinitely large errors ; we may 
consider that in careful observation, errors which transgress 
by the whole amount of the value to be measured, do not 
occur at all; the curve will then remain concave and cut 
the axis at two points. 

I cannot here go into the lengthy investigations which 
have been instituted with a view to determine more exactly 
the most probable form of these curves, their equation and 
from that the chance that an individual error will occur. 
Still I should like to give my reader some idea of the 
means to this end ultimately employed in such speculations, 
though I will not follow them more closely than my purpose 
requires. I will at once drop out of sight the tracts of the 
curve which approach the axis of abscissae; we are only 
concerned to ascertain the chance of such errors as we 
must expect to fall into even in careful observations, and 
accordingly we shall only consider a short arc of the line, 
which lies on either side of the apex. We have seen that 
this line cannot be a straight line ; the next simplest thing 
to assume is that its equation is of the second degree. The 
symmetrical values of the ordinates on this side and that of 
the zero-point are possible on such an assumption, accord- 
ingly we make it and choose from among the sections of 
the cone, which however all lend themselves to the experi- 
ment, — the circle. Let the longest ordinate drawn at the 
zero-point of the abscissae, denote at once the true value of 
the quantity to be measured and the magnitude of the 
probability that this true measurement has been obtained 
in our observations. Let the abscissae ±a, ±ß, ±y 
denote the size of the errors by which the different 
measurements diverge from the true value r ; for the 
present we think of these as expressed in parts of this true 
value, so that + a + ß . . . are for r = 1 proper fractions of 
unity, while if we take r — r they must be replaced by 
+ ra 9 ±rß, . . ; finally the ordinate y which corresponds to 

Logic, Vol. II. L 



1 46 DE TERMINA TION OF INDIVIDUAL FA CTS. [Book II. 

each abscissa denotes the chance there is of that particular 
error in measurement being made, which deviates from the 
true r by the size of this abscissa : if then we assume that 
the equation of the circle holds for the curve in question, 
y is — V i—x\ where x is the general expression for the 
changing values «, /3, y. Now we saw that the chance that 
different and mutually independent events will concur is 
measured by the product of their respective chances. 
Bearing this in mind we shall see that where, as here, we 
are directly compelled to assume a number of errors in our 
measurements, because these do not harmonise with an 
ulterior condition, and where, moreover, various combina- 
tions of errors may be assumed, which would all satisfy this 
condition, that combination of errors is the most likely and 
should be assumed, which allows to the product of the 
individual chances of error the highest value. Now this 
product consists of nothing but factors of the form 
rVi—x 2 and it clearly reaches the highest value, when all 
the several factors take on at the same time the highest 
values compatible with the conditions of the problem. This 
happens, when in all factors at once the subtractive elements 
— here the sum a 2 + /3 2 + y 2 — are reduced to a minimum. 
This minimum value presupposes, as one easily finds out, 
that the sum of the errors a + ß + y . . . = o ; and this can 
only be the case, when these first powers of the errors have 
different signs, and must be the case when the arithmetical 
mean of the observations to which they belong, is taken to 
be the true value r of the quantity to be measured. We 
thus find that this obvious and in simple cases satisfactory 
principle falls within the lines of the process of determining 
r by means of the sum of the squares of the errors. Now 
suppose we have made m observations of a quantity and 
have derived different arithmetical means from them, by 
crediting them each with this or that error and correcting 
it accordingly; our method seeks to determine .that par- 
ticular mean which comes nearest to the truth, inasmuch 



Chap. IX.] METHOD OF THE LEAST SQUARES. 147 



as it rests on the most probable combination of such 
corrections. 

In the above we have not tried to exhaust the subject, we 
have only just approached it in such a way as to give a 
general idea of what is meant by this method of the least 
squares and of how it came to be called by that name. Our 
brief exposition will not serve as a basis for a number of 
more delicate solutions, in regard to which as well as in 
regard to [the introduction of the calculus the reader must 
be referred to the classical exposition of Gauss and to the 
text-books which found upon that exposition. It must be 
borne in mind that the validity of the method always 
depends on certain very probable though not strictly 
demonstrable presuppositions; for its full and adequate 
substantiation we must look to the results to which, 
especially in astronomy, it has led. 



l 2 



CHAPTER X. 

Of Elections and Voting, 

289. Elections and voting are processes of framing 
judgments ; judgments, that is, whose validity we mean to 
create by our own decision, and not merely to acknowledge. 
The logical calculus has taken account of these processes in 
various ways. It has been asked what expectation of a just 
verdict, or of a proper decision, or of a wise election, can 
be based upon different forms of procedure; but, as 
questions like these can never be answered apart from 
special and arbitrary presuppositions of a psychological 
kind, I shall here exclude them, and confine myself to the 
enquiry into the means of attaining what is formally the 
object of all voting, namely, a decision that shall express as 
completely as possible the collective will of the voters, 
independently of the degree of wisdom that may guide the 
several wills which go to compose it. 

In common life such a collective will takes the shape of 
public opinion ; and the matter which it affirms or rejects 
has been gradually defined by the countless reciprocal 
influences of all who have the power of manifesting incli- 
nation or aversion. But a logical treatment presupposes 
that the matter in question is already put into shape as a 
definite proposal For a series of proposals V, IV, Z; that the 
expression of will takes place by simple acceptance or re- 
jection of what is so put forward ; and finally that there is a 
definite and limited number £ of equal votes, to which and 
to no others it belongs to establish the collective will. 



ABSOLUTE MAJORITY. 



149 



290. To take the simplest case : if there is a single 
proposal Fput forward, and a decision absolutely must be 
arrived at, the only possible ground of determination is an 
Absolute Majority. It is the only result which cannot help 
occurring either for or against V\ supposing the case of 
equality of votes to be provided for by some fixed agree- 
ment as to a casting vote, or as to giving the preference to 
the affirmative or to the negative conclusion. But there 
are great limitations on regarding an absolute majority as 
the true expression of what could rightly be called the 
collective will of the voters. The several votes are them- 
selves no exhaustive expression of the several wills ; being 
restricted to 'Yes' or 'No' they have no means of dis- 
tinguishing a decided will for or against from mere accept- 
ance or non-resistance. This constant defect in all voting 
can only be remedied by previous Discussion. This allows 
fitting expression to different intensities of affirmation or 
rejection, and gives scope for the influence of personal 
authority, which has to lose its power in the actual and 
formal voting in which the votes must be counted and not 
weighed. It is for the individual's sense of propriety to 
decide how far in the subsequent giving of the votes 
account is to be taken of the division of feeling for and 
against, which after discussion is at least known to all. 
Other conventional rules, such as the requirement of a 
two-thirds' majority, diminish this evil without removing it ; 
the only unambiguous result would be unanimity, but neither 
it nor the two-thirds' majority can be required without 
endangering the certainty of coming to a decision. So 
these two regulations are only appropriate where there are 
other weighty reasons for giving to the conservative pre- 
ference for the existing state which is known, an advantage 
over the impulse to innovations whose result is unknown. 

291. No general reason can be found in Logic for 
departing from an equal value for all participating votes; 
but in actual life there have been both fair and unfair 



150 OF ELECTIONS AND VOTING, [Book II. 



reasons constantly operative to attach different weights to 
the votes, so as to give an advantage whether to greater 
wisdom, or to the more important or the more specially 
menaced interest, or to claims to peculiar preference which 
had some historical origin. It is sometimes done by 
simply counting the single vote of the preferred person as 
equal to several votes; sometimes by breaking up the 
totality of voters into a number of groups in each of which 
a separate vote is taken, and substituting the majority of 
the majorities which arise in this voting for the absolute 
majority of the total number ; sometimes by having recourse 
to indirect voting, in which each of the groups transfers its 
right to a delegate and leaves the decision to the majority 
of these nominees. 

The first case requires no separate consideration; the 
last withdraws itself from all logical treatment in cases 
where the deputy so commissioned has to vote indepen- 
dently himself, and not to represent a decision already 
taken by his electors. For the certainty with which the 
result in that case corresponds to the collective will de- 
pends upon the doubtful reliability of the electors' judgment 
in estimating the agreement between their deputy's senti- 
ments and their own. On the other hand, the second case, 
that of division into groups which are to vote separately, 
has the following determinable peculiarities. 

i. If we take the total number S of the votes 
as = 2 m . 2 n, one of these factors indicating the number 
of groups made, and the other the number of votes in each 
group, then (m+i)(n + i) will be the number of votes in 
the absolute majority of the several absolute majorities 
which result within these groups. And this value remains 
the same if we substitute for one or both of such even 
factors the odd numbers next above (2 m+ 1) (2 n+ 1). 
Suppose M on the other hand to be the simple absolute 
majority of the total number of voters S when voting with- 
out subdivision, we may easily convince ourselves that 



Chap.X.] THE 1 WINNING MINORITY' 151 

(m + i)(n+i) is less than M for all uneven S which are 
greater than 7 and for all even £ which are greater 
than 12, and so in all cases that need be considered 
with reference to voting. Thus it is always possible by a 
suitable subdivision of to bring about a decision resting 
on the minority of the total number of voters ; and it may 
be asked which modes of division are the most adapted for 
making this winning minority as small as possible. A 
precise answer to the question would be far more lengthy 
than the matter deserves ; for in application we shall always 
have to be satisfied with an approximation, because our 
precise estimate would be made useless by any trifling 
accident that prevented a vote from being given on which 
we had reckoned. So I content myself with what follows. 

2. If we consider S as the product of two even or of two 
uneven factors, and thus either 

= 2^.2« or = (2 m + 1) (2 n + 1), 
if we replace m in the formula for the winning minority by 
an expression in terms of n and S, and if we differentiate 
with respect to n, we obtain as condition of a minimum 
2?ior2n + i = VS which gives the other factor s = V S, 
and therefore m = n. 

If we take £ as product of an even and an uneven factor, 
= 2 m (2 n + 1), we obtain in the same w T ay, as a condition 
of a minimum that the even factor = V 2 S, which gives 
the uneven = V \ S. The manner of their deduction pre- 
vents either of the formulae from applying precisely in these 
cases where both the number of votes and that of groups 
can only increase by entire units, and not continuously; in 
particular, their application cannot be regular for small 
numbers of whose amount a unit is a considerable fraction ; 
and lastly, the advantage of odd numbers over even, as the 
winning minority for {2 m -\- 1) {2 n + 1) is not larger than 
that for 2 m . 2 ^, will also be detrimental to the influence 
of these rules. Still, for high values of S y as unity, or the 
difference between odd and even, forms a progressively 



152 OF ELECTIONS AND VOTING. [Book II. 

smaller fraction of their amount, these two formulae really 
give the two least values of the required minorities. They 
are obtained by separating S into two factors either as 
nearly as possible equal to each other and to the square 
root of S, or one of them as exactly as possible double the 
other. Thus 225 considered as 15.15 and as 9.25 gives 
the two least minorities 64 and 65, but as 5 . 45 and 3 . 75 
the larger ones 69 and 76; 11025 as 105. 105 and as 
147.75 tne l east minorities 2809 and 2812, while on the 
other hand as 175 . 63 and as 9 . 1225 it gives the larger 
numbers 2992 and 3065 ; and finally 20,000 breaks up to 
best advantage as 200. 100 and as 125. 160, with the 
minorities 5 151 and 5103. In the case of small numbers 
the influences of the different conditions cross each other 
very markedly; 36 taken as 6 . 6 gives the minority 16, but 
even as 4 . 9 gives the smaller minority 15 because of the 
favourable influence of the odd factor; the most advan- 
tageous subdivision is 3.12 giving 14 as the minority; for 
in this the even factor 1 2 comes nearer to the square root of 
2 S=J2 (which is greater than 8) than does the even factor 4 
in the division 4 . 9. On the other hand 81, being the square 
of an uneven factor, has no subdivision more favourable 
than 9 . 9 giving 25 for the minority; the other into 3.27 is 
too remote from both conditions. For 144 one minimum 
49 is obtained out of 12 . 12, the other 45 out of 9 . 16. 

3. In the first of the most favourable cases, that of 
equal factors, the winning minority, expressed in terms of S, 
— (1 + -J VS) 2 ; in the second, that in which one factor is 
double the other, it == 

(i+iv^s) (i+^yp). 

Both expressions approximate to the value \ S (only the 
second does so more slowly) as S becomes greater, but 
remain always greater than that fraction as long as S does 
not become infinite. Thus the winning minority has a 
lower limit, and can never, by the most advantageous sub- 
division, be reduced to a quarter of the total number of votes. 



Chap. X.] 



SUCCESSIVE VOTINGS. 



153 



4. Finally, S may be a prime number, which in any case 
can only be made divisible if increased or diminished by a 
single unit at least : i. e. for the present purpose, by giving 
to one of the groups a single vote more or less than the 
others have. A choice is therefore inevitable, and may be 
exercised as we please ; beyond a doubt it is equally justifi- 
able to consider 67 as 66 + 1 and as 68 — 1, making in the 
first case 5 classes of n votes and 1 of 12, in the second 3 
classes of 17 and 1 of 16 ; if it is required for the sake of 
fairness that the majorities which make up the winning 
minority shall always include those of the more numerous 
classes, we shall obtain in the first case 3.6 + 1.7 = 25, in 
the second 3. 9 = 27. When once this path is open, it is 
followed even where there is no need ; and then the in- 
equality of groups is readily put up with as long as it 
remains within reasonable limits ; and it further diminishes 
the winning minorities very considerably. Thus we obtain 
for 64 = 6 . 9 + 1 . 10 the minority 3.54-1.6 = 21 (even if 
we make a rule of requiring the majority of the larger group 
to be in it) while 8 . 8 only gave the greater minority 25. 
We all know that this resource has been abundantly em- 
ployed from Servius Tullius downwards to a very unfair 
extent, which must look for its justification not to Logic, 
but to politics. 

292. When we come to choosing between different pro- 
posals V, IV] Z, Logic as such would make requirements 
that diverge from the usages observed in practice. If a 
number of persons desire to unite in a collective determina- 
tion such as to produce the greatest general satisfaction, 
they ought not to obtain the result as an inevitable conse- 
quence of a summation of declarations of will, none of 
which takes account of any of the others. A rational will 
must attach importance to not giving its decision without 
knowledge and consideration of the other voters' inclina- 
tions or aversions, opposed to its own ; especially as the 
necessity of finally declaring itself in a bare Yes or No leaves 



154 OF ELECTION AND VOTING. [Book II. 

no means of finding expression for the different intensities 
of its volition and so of securing for it neither more nor less 
than the just measure of effect. Previous discussion, to 
which I referred above, cannot entirely satisfy this require- 
ment ; for if everyone wanted to declare himself completely, 
the discussion would turn into voting, only without the 
precision of form which makes the ascertainment of the 
final result easy and certain. So an attempt must be made 
to effect by the actual mode of voting as nearly as possible 
what discussion aims at doing. 

If we consider V, IV, Z, as three persons, one of whom 
is to be elected, we may adopt the following procedure. A 
preliminary vote upon all three candidates at once would 
show what degree of approval each of them meets with, 
compared to the others. If no one of them obtained an 
absolute majority of the votes, the relative majority could 
not be taken as decisive except in elections of very small 
moment ; but it has an importance which may be seen in 
everyday life. The candidate who has the most votes 
compared with the others attracts attention and often gains 
the other votes as well ; but just as often his prominence 
arouses antagonism, and compels his opponents to combine 
in support of a rival. Hence it is the general rule to require 
an absolute majority ; it gives the only security that the sum 
of negative votes must be less than that of affirmatives, and 
that therefore the will of the majority has been hit upon ; 
and this is the ultima ratio which must always give the final 
decision when opinions remain unreconcilable and yet a 
collective resolution is indispensable. 

But further ; if an absolute majority has been obtained 1 
for one of the candidates, say for V, still it is neither 
essential, nor right in itself, to accept this at once as the 
decision. This preliminary voting only showed the number 
of voters who preferred or postponed each of the candidates 
to the others ; the degree of such preference was left 

1 [I.e. in the preliminary vote.] 



Chap.X.] UNEQUAL VALUE OF VOTES, 155 



undefined, and so also was the feeling of each voter to the 
candidate whom he has not named. To bring this to light 
a second and tripartite vote would be required; being a vote 
of Yes or No upon each of the candidates separately so as 
to give each elector the possibility of directly recording his 
vote against any particular candidate, while before he could 
only express it by preferring another to him. If we suppose 
that in the first voting V obtained n votes out of 20, W 5, 
and Z 4, then, excluding incomprehensible inconsistency on 
the electors' part, every candidate will retain in the second 
process the votes of those who preferred him to the other 
two in the first, but the remaining votes may be very 
variously divided. It is possible that V may now meet with 
a decided opposition of 9 votes, while Z who was only pre- 
ferred by 4 may find no opposition at all and win 16 votes 
more; and W may get 10 of the 15 which he had not 
before. 

To obtain a final result out of this it must be considered 
that the votes obtained in these different votings are of 
unequal value. Those of the first process showed how 
many voters thought a candidate the best, and though their 
approval may have been very different in degree still we 
may regard all these preference votes as homogeneous and 
attach the same weight m to all of them. For the best that 
anyone can say of a candidate with reference to the election 
as such, is that he is for him ; whether he respects him 
more or less apart from this is indifferent, for every election 
can only aim at the best result under the given conditions, 
and at not the best absolutely ; any one who is for V or W 
under the given conditions is for him altogether. The same 
assumption holds of the negative votes in the second voting; 
anyone who has the opportunity of pronouncing directly on 
V ox IV by Yes or No, and votes against both, is absolutely 
against them, and has accomplished his will completely, as 
regards this election, if the vote of rejection is carried ; no 
matter how thoroughly he may hate or despise V or W in 



OF ELECTIONS AND VOTING. [Book II. 



other ways. So all the unfavourable votes may be con- 
sidered as homogeneous and assigned the same weight q. 
But those favourable votes which are only obtained in the 
second voting are obviously of less value than those which 
were obtained in the first ; they are only permissive, while 
the others were preference votes ; and this difference, which 
marks a middle grade between voting for and voting against, 
is of course of importance for the election in hand. Still, 
what weight a permissive vote ought to have compared with 
a vote of preference, is a question which the voter who gave 
it could probably not answer with precision ; besides, his 
acquiescence would not indicate the same degree of approval 
for every candidate to whom he gave it, but a greater, 
perhaps, for W than for Z. Therefore, though it at once 
involves a serious failure of accuracy, yet the only possible 
attempt to make a general estimate even approximately of 
the difference of the permissive votes from the preferential 
is to assign to all votes of the former class a common value 
which must be a proper fraction of m y and the amount of 
which can only be fixed by convention. On these assump- 
tions the votes would be calculated in the above example as 
follows : for V, 1 1 m — 9 for W, 5 m + 1 o p — 5 q, for Z, 
4^+16/; and so finally, if we arbitrarily take m—q (giving 
the preferential vote equal weight with the vote of rejection) 
vi 

and P — ~ (giving the permissive half the weight of the 

preferential), the result would be only 2 votes for V, on the 
other hand 5 for W and lastly 1 2 for Z, in marked contrast 
to the result of the first voting. 

Various circumstances combine to make these logical 
requirements unrealisable in practice. In the first place, on 
grounds of social propriety, we should desire to avoid voting 
against persons altogether. Next, even if it were admitted, 
there would be great reason to doubt whether the second 
voting would be carried out with the requisite impartiality, 
even supposing it to take place before the first. Those who 



Chap.X.] AFFIRMATIVE AND NEGATIVE VOTES. 157 

were determined to give their votes of preference to V would 
probably not admit even to themselves that they could be 
content with W or Z, and their 1 1 votes would appear in 
the permissive voting too as so many votes against Wand 
against Z as well. And lastly there would in every case be 
the same preliminary question what was best to aim at with 
a view to the matter in hand, whether the completest satis- 
faction of the majority or the greatest average satisfaction of 
all ; and this would have to be determined before the ratio 
of weight between affirmative and negative votes could be 
fixed. It need not necessarily be equality; on the contrary, 
there may be cases in which a single unfavourable vote may 
fairly balance more than a single favourable one, and the 
decision would have to be obtained not so much by the 
greatest number of votes for a candidate, as by the lowest 
number of votes against him. It clearly makes a difference 
whether the matter in hand is the decision of some gravely 
responsible business, say an election to some office of 
political importance, or whether it is the organisation of 
amusements in common, perhaps the election of the president 
of some social gathering. In the latter case it would be 
absurd to make 9 members out of 20 discontented in order 
to give 1 1 others complete satisfaction ; but in the former it 
may be reasonable to satisfy the majority of decided wills 
completely, rather than light upon a choice that only met 
with the lukewarm approval of all. But it is just in the 
second case, where the method described would give the 
most desirable result, that the inadmissibility of negative 
votes makes its application difficult ; in the first, where its 
result might be less desirable, its application would be less 
difficult, for in this case the votes of rejection would be a 
less serious slight, as they might possibly be directed against 
the views represented by a candidate and not against him 
personally. 

293. There is another mode, a sort of process of elimina- 
tion, that may answer our wishes when an election has to be 



158 OF ELECTIONS AND VOTING. [Book II. 

made out of a very large number of candidates ; as when a 
constituency has to name some one out of its own number. 
It is usual in these cases to take a first vote by way of fixing 
upon some three names which are the first to attract the 
electors' attention before minor considerations come in to 
restrict it, and at present therefore appear the most desirable 
to each of the voters. It is possible in this process to 
attach distinctions to the order in which each elector names 
or writes his three candidates ; and to put the one mentioned 
first above the others; but I assume for simplicity's sake, 
that the order of naming is quite indifferent. Then it is 
conceivable, though very improbable, that the same three 
candidates, V, W, Z, may receive all the votes ; if this 
occurs, it becomes impossible for a final decision to be 
obtained by election, for a fresh vote could not give a 
majority for any of the three unless some voters retracted 
their previous decision without any ground in circumstances 
for doing so. In this and all similar cases the only re- 
maining possibility is either the lot, or the decision of some 
external will, e.g. that of a higher authority. 

On the other hand, if Fand no one else obtains a vote 
from every voter, his election is decided beyond a doubt, 
whatever number of votes W and Z may have obtained \ 
for then there are no concealed votes of rejection which 
nothing but want of opportunity has hindered from being 
recorded. But supposing V to have only obtained an abso- 
lute majority, Wand. Z considerable minorities, and the rest 
of the votes to have been scattered, then it becomes possible 
that there are such votes. So, considering our observations 
above, we cannot hold it quite justifiable to break off the 
election at this point and regard V as elected ; it is better 
to take a fresh vote for W against Z, so that actually voting 
against W may be avoided by voting for Z, and vice versa. 
One of the two must obtain a larger or smaller absolute 
majority. Supposing W is successful, a third vote as be- 
tween him and Fwill give a final decision. Of course this 



Chap. X ] 



ORDER OF PROPOSALS. 



*59 



final voting will be wholly superfluous if the absolute majo- 
rity which was in favour of V on the first vote maintains 
itself unaltered ; but a reasonable motive for a change of 
feeling may have been furnished by the result of the second 
voting. If in it the votes for IV and for Zare nearly equal, 
it would prove either that V's opponents are not united, or 
that no other choice meets with more uniform agreement 
than that of V, and this would give the previous majority a 
reason for persisting in their conviction. If on the other 
hand W got all the votes, the majority in question might 
think this a good reason for going over in the last vote to 
the minority, already considerable, in favour of W, in order 
to produce a result which should have no decided opponents. 
Many more modifications are conceivable ; I will not follow 
them out, for the discussion threatens to be longer than its 
importance warrants; moreover it is at least doubtful 
whether such a process of elimination is really more flatter- 
ing than open rejection. And finally, V, who is elected, 
may decline the post. This alters the conditions with re- 
ference to which the votes were given so completely, that it 
becomes necessary to repeat the whole process of election, 
or perhaps to make an independent selection of Y, as a third 
candidate besides Wdcnd Z. 

294. If V) and Z, are not persons, but legislative 
proposals, there is no reason for shunning the direct nega- 
tive vote ; and it might be demanded on logical grounds 
that a vote of Yes or No should be taken on each of the 
measures proposed, but that the obtaining of an absolute 
majority by one should be no bar to voting on the others. 
The decision would then depend either on the largest of the 
majorities or on a fresh vote which would be final. This 
procedure would cause those whose opinion had gained a 
considerable number of votes, to adhere to it in the final 
voting; but any who found theirs in a hopeless minority 
would have time to attach themselves in the final vote to the 
opinion which they liked next best to their own and which 



i6o OF ELECTIONS AND VOTING. [Book II. 

might have a chance of gaining the deciding majority by 
their accession. Still, there is here too the same psycho- 
logical reason which I mentioned before, against this pro- 
cedure; any one who decidedly preferred a proposal V 
would not declare freely that W or Z would also be tolerable 
to him, but would be tempted to reject them both. There- 
fore as it is the traditional rule that the adoption of one 
proposal ipso facto shelves all posterior ones, the order in 
which VWZ are put to the vote acquires great importance 
as affecting the choice made. I concur with Trendelen- 
burg's statement 1 of the wish that may be felt on logical 
grounds respecting this arrangement, the most difficult pro- 
blem of parliamentary tact ; viz. that the adherents of every 
opinion should have an opportunity of emphasising it with 
all the weight they can command ; of negativing what they 
want to reject directly, not indirectly by the acceptance of 
something else which has only their partial approval ; of 
affirming what they wish immediately and exclusively, not 
by the rejection of something else of which they disapprove 
only in part ; and finally, that it should be possible for 
everyone to begin by defending and recommending what 
he thinks best of all, and only to retire upon his second or 
third best after the first has failed. But whether the uni- 
versal accomplishment of this postulate for everyone entitled 
to vote and in respect of every proposal before a meeting is 
not as a whole frustrated by a fundamental contradiction ; 
whether, therefore, it is conceivable that everyone's senti- 
ments should be gratified by just those proposals being 
broken up on the parts of which he thinks differently, and 
just those united which he wants to see accepted or rejected 
together, demands no investigation. It is quite clear that 
in each case the solution of the problem can only be ap- 
proached by an acumen developed in long and uniform 
practice, after entering thoroughly into the subject-matter 
under discussion. The procedure to be observed can only 
1 [' Ueber die Methode der Abstimmungen/ Berlin, 1S50.] 



Chap. X.] THE 6 ORDER OF THE DAY! 161 

be learned or taught by help of definite instances, not by 
universal symbols representing possible cases, and only in 
practice ; general rules can give very little help. 

295. It may happen to begin with that the proposals in 
question VWZAo not compose the complete disjunction 
between the members of which there is a choice, but that 
there is a fourth member consisting of the rejection of one 
and all of them ; i.e. that speaking generally a new resolu- 
tion is as such unnecessary, and it is possible to maintain 
the status quo. There are two reasons that may lead to the 
choice of this alternative ; either the desire to protect that 
particular status quo on principle against all innovation, or 
the absence of an acceptable proposal among those put for- 
ward, though on grounds of principle there is no opposition 
to a reform. It is important to provide expression for the 
distinction between these two dispositions. The mere re- 
jection of all individual proposals successively does not 
provide it ; this only proves that the change which would 
have been acceptable has not been proposed ; but it should 
be possible to reject generally and as such the invitation to 
change which all the proposals have in common. This is 
effected by the motion to pass to the order of the day, that 
is, therefore, that the whole of the proposals in question 
should be excluded from being debated or voted on, and so 
their common element should be negatived in a correspond- 
ingly general form. Where anyone's will is in favour of 
such a negative it is his parliamentary duty to contribute to 
a full expression of the state of opinion by making this 
motion, and not to content himself with throwing out all 
separate proposals, until it has been rejected. Even where 
there is only a single proposal instead of several, the motion 
of the order of the day may be in place ; its meaning then 
is to reject not this particular proposal as such, but the 
general intention out of which it has arisen and others 
might arise. Thus the order of the day when voted without 
a statement of grounds, may act as an expression of con- 

Logic, Vol. II. M 



i62 OF ELECTIONS AND VOTING, [Book II. 



tempt for a proposal that is legally or morally disgraceful, 
or as a refusal to entertain one that is extraneous, and 
beyond the competence of the voters, or finally as a rid- 
dance of a dangerous proposal the mere discussion of which 
it is the interest of the commonwealth to avoid. Or such 
acts of rejection may be mitigated by a motived order of the 
day, which recognises in its statement of grounds what there 
is in the proposed measure that is just in itself, but denies 
the propriety of introducing and debating it at that particular 
moment. 

296. If two proposals Fand W are so related to each 
other by way of subordination that Wis an 4 ameliorative 
motion ' or amendment that aims at modifying the purport 
of the substantive motion V by addition, omission, or 
alteration, then it is logically speaking a correct usage to 
take a provisional vote on the amendment before the 
final vote on the substantive motion. For no de- 
cision on this latter can rationally be solicited from the 
voters till its wording is completely and unequivocally 
settled ; certainly not while its details are open to subse- 
quent modifications the acceptance or rejection of which if 
it could have been foreknown might well have succeeded in 
totally reversing the favourable or unfavourable impressions 
which had been prematurely recorded. The vote on the 
amendment W serves to fix unequivocally the purport 
which the substantive motion V has when put to the vote ; 
therefore the rejection of /^annuls the previous adoption of 
the amendment, which was only provisional. 

If there are several mutually exclusive amendments W 
and Z to a substantive motion V, or several accessory pro- 
posals about the special modifications necessary to applying 
V in practice (as often happens when details of quantity 
remain to be fixed), the safest course would be to vote 
separately on all such proposals and let the decision go by 
the greatest majority. Only if, according to the usual prac- 
tice, the acceptance of one by an absolute majority is to 



Chap.X.] PRIORITY OF MOTIONS. 163 

exclude all the others from being voted on, the order of 
putting them to the vote becomes important, and the 
obvious advice is to arrange the proposals so that the two 
least divergent shall always be next each other. This is the 
practice under a rather different form in both kinds of auc- 
tion, that by bidding and that by offering 1 , and in these 
cases people actually calculate, and quite fairly, on the un- 
certainty in which each person who bids must be about the 
degree of the others' desire. For as the bid or its accept- 
ance 1 by the buyer are alike voluntary, the customer is 
merely declaring what value the object has for him accord- 
ing to his own estimate and no right of his is attacked by 
the open competition of others or by his ignorance of the 
absence of any eager desires but his own. Offering seems, 
speaking generally, in favour of the seller, for it compels 
the buyer to take the object at the highest price which he 
thinks he can afford to give, though he would give less if he 
could foresee the absence of competition; bidding is in 
favour of the buyer because if there is such absence it is 
available to him ; if not, at least he has only to advance on 
the last bid, and his time for decision is not excessively 
curtailed. This is a procedure in which an individual tries 
by a contest with others to secure a legitimate personal 
advantage ; so its analogy is not in spirit very appropriate 
to the efforts of a multitude to bring to pass by common 
action a resolution advantageous to the common weal. 
Still, in form, they must take pattern from the procedure of 
' offering/ only, as a rule, it will be rare to find proposals 
that can be so simply arranged in a quantitatively graduated 
series; most commonly W Z . . . will differ in purport so as 
to be hard to classify. In that case they must be arranged 
according to their anticipated degree of conformity to the 
general will ; those that are furthest removed from the status 
quo, that demand the most extraordinary and ample mea- 

1 [I. e. in a ' Dutch auction,' where the auctioneer offers successively 
lower prices, and the first customer who accepts is the buyer.] 

M 2 



164 



OF ELECTIONS AND VOTING. 



[Book II. 



sures, and therefore have little probability of success, would 
have a claim to be put to the vote first ; in order that, if, 
contrary to expectation, they were destined to turn out in 
conformity with the general will, the expression of that will 
might not have been made impossible by starting from a 
more likely proposal which might easily unite all the votes 
with premature resignation based on that erroneous estimate 
itself. After the rejection of such extreme proposals we 
might pass, as in the mathematical method of limitation, to 
the mean terms of the series which are more likely in them- 
selves ; with the aim of procuring the final decision in 
favour of the proposal which involved the least possible de- 
viation from general satisfaction. But all these rules are 
ultimately inadequate; for instance, where the business is 
to decide upon a composite whole whose different parts can 
only be discussed by degrees, it must always be impossible 
while the special deliberations are still proceeding to dis- 
cover all the inconsistencies, inconsecutivenesses, and contra- 
dictions which would arise from the ultimate conjunction of 
all the details of the plan, perhaps variously modified. In 
such cases, as in the case of amendments, the special de- 
liberations should be treated as no more than preliminary, 
and an assembly should reserve to itself the power, by a 
vote on a second reading or by some final vote, to anni- 
hilate the monster which its united efforts have brought into 
existence. Lastly, it is true that the formal aim of all voting, 
to arrive at a collective will, would involve in the first place 
the establishment of a decision Z such as to give all mem- 
bers of the society the greatest attainable average satisfaction 
M, considering the lesser satisfaction of one as compensated 
by the greater satisfaction of another. At the same time it 
is also to be desired that in order to the performance of the 
obligation arising from the acceptance of Z an equal com- 
pliance M were to be reckoned upon in all members. I 
have explained why the former end cannot be attained in 
perfection. The latter on the other hand is of course a 



Chap. X.] 



DUTY OF COMPLIANCE. 



165 



desire unrealisable by logical means \ only this may be de- 
duced as a logical rule which the nature of ethical ends 
makes necessary to their realisation, that here (for Logic 
can require it nowhere else) we should subordinate our 
personal conviction to the general opinion when different 
from our own. 



BOOK III. 

ON KNOWLEDGE (METHODOLOGY). 



INTRODUCTION. 

In the enquiry instituted in the last book as to the means 
by which we are enabled to arrange the manifold content 
presented to our minds under those ideal forms of appre- 
hension and connexion with which we became acquainted in 
the previous book upon Pure Logic, nothing was said as to 
the general question of logical methods with which a theory 
of the nature of thought ordinarily concludes. In passing 
over the subject in that place I conceive that I was guilty 
of no unpardonable omission, nor was it mere caprice which 
led me to reserve that and other questions akin to it for this 
concluding section of my work. 

297. Since the time of Aristotle philosophers have dis- 
tinguished between the analytic and synthetic methods, 
from points of view essentially the same in all cases, re- 
garding them as the two ultimate forms to which all 
methods of scientific procedure which the movement of 
thought follows may be reduced. In the view of antiquity 
any subject-matter presented for scientific investigation was 
to be submitted to a process of dissection which should 
trace it to its simplest elements or to its most universal 
conditions. Thus the analytic method was a retrogressive 
operation proceeding a principiatis ad pri?icipia^ while the 
principles when discovered formed as it were the blocks 



INTRODUCTION. 



167 



which the synthetic or progressive method proceeded to 
build up into the individual objects of experience. 

The two expressions analytic and synthetic no longer 
precisely answer to the instincts of modern speech, and we 
might easily be tempted to interchange their meanings. 
We no longer indulge the hope that a mere dissection of 
the object presented can discover within it the principles 
of which we are in search ; experience has taught us on the 
contrary that for the human intelligence general principles 
have largely to be created out of a combination and com- 
parison of the manifold facts of experience, and they appear 
to us therefore when we have arrived at them, as the final 
outcome of a synthetic operation of thought. In the same 
way we are no longer disposed to limit ourselves to a point 
of view which regards general principles as atoms of truth, 
from the mere piecing together of which particular truths 
are derived ; to us, rightly or wrongly, general principles 
appear rather as containing within themselves a capacity 
of development, and we regard the derivation of the con- 
ditioned from its conditions as consisting at least as often in 
an analysis of the content of the conditions as in the com- 
bination of them. 

But the question of language is not worth debating further, 
for it is plain to begin with that neither of the two methods 
can be carried out, at all events in any general application, 
without the other. No method of analysis can arrive by the 
mere dissection of the particular object presented to it, at a 
principle or a general truth, unless at every stage it compares 
the result a of the last step with some general proposition T, 
and by endeavouring to bring it under the latter — that is to 
say at this point by an act of synthesis — makes sure whether 
a is an ultimate principle, or whether it may not involve 
some contradiction for the removal of which it may be 
necessary to continue the analysis further in one direction 
or another. Nor need the proposition T which has to be 
recognised, in such a process by any means always belong 



1 68 



INTRODUCTION. 



[Book III. 



to those formal logical laws whose supremacy over the 
modus procedendi of every conceivable method is acknow- 
ledged as a matter of course. On the contrary it must 
often, if any real advance is to be made, be a concrete 
proposition which Logic cannot furnish, but which it has 
to accept on purely extralogical grounds, and to which 
the results arrived at by the analysis have to be sub- 
ordinated. 

As little can a method of synthesis make any way without 
the help of analysis. Even supposing it to start with a 
number of elementary truths A, jB, C in its hand, it could 
never get beyond the tautological proposition that these 
several truths are all true at once, unless it can go on to 
show how by their possessing simultaneous validity in 
respect of one and the same object this or that fresh con- 
sequence x or y is necessarily developed. But w r hether 
x or y will follow can only be decided after an examination 
of the nature of the object in question, that is to say by a 
return for the time being to analysis. Such an analysis can 
alone furnish us with the determinate minor premiss which 
we require to combine with the general truths w r ith which 
we began, and which supply our major premiss, in order 
to proceed once more synthetically to a determinate con- 
clusion. 

It must be allowed that in certain departments the 
synthetic method has the appearance of a greater indepen- 
dence. Thus geometry is able to create the objects to 
which it desires to apply its general truths, as it goes along, 
and the analytic statement of the data which are accepted 
for the purpose of deducing each new proposition occupies 
only a small place in its demonstrations. Still in reality it 
cannot be wholly absent. And in the *wider provinces of 
scientific activity which are concerned with the synthetic 
construction of real things, the progressive movement from 
principles to facts is always preceded by a retrogressive 
operation, consisting in a comprehensive analysis of the 



Book III.] 



INTRODUCTION. 



169 



data, which is essential to determining for the synthetic 
procedure itself the directions in which it has to look for 
those minor premises with which its general principles 
cannot possibly dispense. 

298. Thus in fact the distinction between the two methods 
runs up practically into the following antithesis, which has 
long attracted attention : — the Analytic method is essentially 
the method of investigation, having the discovery of truth 
for its object ; the Synthetic that of exposition, of which the 
object is to exhibit a body of truths, whether obtained in 
one way or in another, by direct or indirect processes, in 
their natural and objective connexion. And by exposition 
I do not merely understand communication to others, a 
purpose for which an exhibition of the subjective process of 
discovery is no less necessary and no less instructive. I 
mean rather the framing of the results arrived at into a 
logical whole, in which form alone they meet the ideal 
requirements which the human intellect makes of any body 
of truth that is to be fixed and independent. 

Such being the case it appeared to me hardly desirable 
to introduce the question of the analytic and synthetic 
methods among the questions of applied logic, inasmuch as 
neither the one nor the other affords any practical contribu- 
tion towards the solution of any definite problems, — the 
analytic method as little as the synthetic, notwithstanding 
that we regard it as the type of all methods of investigation 
and discovery. To instruct a person to employ the analytic 
method is to give him no very helpful counsel ; the cus- 
tomary general definitions of the method really contain 
nothing more than an indication of the direction in which 
the required road still has to be looked for ; to find it we 
must turn to the special expedients of applied logic, in 
using which it makes very little difference whether we choose 
to rank them under the method of synthesis or under that 
of analysis. In the same manner an enquirer who has the 
synthetic method prescribed to him has merely got his 



170 



IN TR OB UC TION. 



[Book ill. 



problem stated : to the question how to solve it, in which 
certainly the rules of a method ought by rights to be of 
service to him, no very sufficient answer is afforded by the 
general direction to work downwards from principles to 
results. 

299. All this is changed if we surrender a certain privilege 
which in the sphere of applied logic we allowed ourselves to 
claim, and endeavour by this surrender to give completeness 
to our treatment of the subject. Thus far in speaking of the 
forms of proof, of the search for grounds of proof, of the 
discovery of laws, we have left our work throughout in a 
certain sense incomplete. Every attempt to establish a pro- 
position went only a few steps back, and came to a stand as 
soon as some other proposition was reached, which served 
for its foundation, and the validity of which was not proved 
but assumed. This procedure answers to the actual course 
of thought in life as well as in the sphere of the special 
sciences. In ordinary life our judgment of things and the 
conclusions we draw concerning them rest not on a single 
proposition T, nor yet upon a clearly defined group of 
homogeneous elementary truths, but on a large number of 
truths of a quite heterogeneous character, yet possessed 
of an equal certainty; here a proposition A, which once 
apprehended forces itself upon us as a necessity of thought, 
there a proposition B expressing an immediate fact of per- 
ception, and presenting itself not exactly as necessary but 
as incontrovertible ; a third, again, C, may be a principle of 
unknown origin, but one which at every moment is being 
put to the proof and confirmed afresh ; finally we have 
many a proposition D springing from equally unknown 
sources, but admitting of no such guarantee of its truth, and 
yet seeming to bear within it an indefeasible claim, a claim 
that is which we believe ourselves bound to satisfy if the 
conceptions by which we bind together the data of ex- 
perience are to answer to the truth. 

Any one of these various points of certainty — and in each 



Book III.] 



INTRODUCTION. 



171 



of them we may suppose a number of elementary convictions 
to be contained and compressed — is used indifferently in the 
living movement of thought as occasion arises, to answer 
any question which comes up for solution ; nay even a pro- 
position which naturally depends on some one particular 
assumption, we often prove from a different one, if its 
dependence on the former is not at once evident. In this 
way we are constantly shifting the bases on which our 
judgments rest ; at one time we set out from some law 
which is evident to us, and determine its effects ; at another 
by repeated observation of the effects we strengthen our 
belief in the law ; consequences which appear to flow by an 
internal necessity from some acknowledged principle, are 
rejected for their improbability as seen from a different 
standpoint ; now we start from A to prove a doubtful B, 
now B appears the more evident of the two, and we use it 
to establish A ; the truth being that whatever possesses for 
us at the moment the strongest psychological certainty 
passes for the point of vantage from which the other more 
wavering beliefs are to be secured. 

300. So entirely unconstrained as this in its operation 
scientific thought certainly is not ; still the actual science 
which we possess, as distinguished from the ideal science, 
which we might wish to possess, has resemblance enough to 
the natural processes of ordinary reflexion. Here also we 
hardly meet with any enquiry into a matter of fact, which 
does not depend for its determination upon certain pre- 
suppositions which we adopt now as undemonstrable but 
certain, now again as undemonstrable and only probable, 
and which are regarded either as ultimate principles of the 
particular science before us, or as vouched for by some 
other science. Even within a single province of enquiry 
the direction in which the required proof is looked for 
varies. Without exactly questioning the certainty of a pro- 
position which we have begun by regarding as the source 
from which others are derived, we come nevertheless to 



172 



INTRODUCTION. 



[Book III. 



believe that there is some other proposition which we may 
place at the head with still greater security, and from which 
we may derive our former first principle with all its conse- 
quences. 

And if we review our knowledge as a whole, distributed 
as it is among the various sciences, we shall find no one of 
these complete and rounded off in itself. In each one of 
them we come upon formal or material principles, whose 
validity is admitted because they are self-evident, or because 
they explain certain facts, while as to their origin and con- 
nexion with each other no enquiry is made if the question 
appears to have no immediate bearing upon the prosecution 
of the science itself. 

This point of view we kept before us while v/e were 
dealing with the subject of applied logic, and we saw no 
reason to depart from it so long as our subject was the 
nature of investigation. For what we commonly call Applied 
Logic, or more properly the exhibition of the possible modes 
in which Logic can be applied, presupposes the existence 
of a variety of cases adapted for its application, and this is 
only possible if the work of investigation consists in taking 
some given fixed point to start from, and then connecting 
this by rule and law with other fixed points which are also 
not proved, but assumed. Such is the character of all 
investigations in which we are accustomed actually to 
engage, and in this our knowledge bears resemblance to 
our life. What was the origin of our race in the first 
beginnings of history we know not, and as little can thought 
carry us to its far-off future ; for most of us the memory of 
those who have gone before fades away in no very distant 
past, and for all alike the prospect granted of the fortunes 
of those who are to come afte r us is still more confined ; 
yet in the midst of this darkness on either hand, there lies 
before us a certain space of life comparatively clear, with 
plain needs, pressing duties, and attainable goals ; our joy 
in existence, and our confidence in acting upon the present, 



Book III.] 



INTRODUCTION. 



173 



are but little impaired by the uncertainty of the beginning 
and the end. And so it is with our knowledge. That there 
is an eternal truth, or a perfect and self-closed circle of 
truths, we do indeed assume, but ordinary reflexion has for 
such a system of truth no form of expression adequate to it, 
nor has it any clear conception how the members of the 
system are related ; single portions only become plain and 
evident to us in a manner which we are incapable of 
analysing to ourselves, in the course of our mind's operation 
as it comes in contact with reality. In the process of 
investigation we resemble men engaged in a narrow inland 
traffic, and we endeavour to connect the uncertain and 
changing scenes about us, with the isolated peaks which rise 
upon our view out of a coherent world of truth which we 
cannot see. 

301. But just as there arrive moments in life when the 
present only seems endurable or intelligible if we can catch 
some glimpse of its connexion with the past and with the 
future, so also in knowledge there are occasions when we 
are tempted to pass out of the petty business of ordinary 
scientific investigation, and reflect upon the points it starts 
from, and the points it aims at ; to ask where they are 
situated, how they are connected with each other, and 
whether they are secure. For the principles on which the 
several sciences repose do not restrict themselves to an un- 
aggressive sway each within its own separate province. 
We need only point to the very different consequences in 
respect of the powers that shape human life which are 
deduced from the principles of mechanical science on the 
one hand, and the deliverances of conscience on the other, 
to see in a single emphatic example how the claims of 
different sources of truth may clash and conflict in dealing 
with a common subject-matter. But even within the field 
of purely theoretical science we may find inducements 
enough to make that which in the living processes of 
thought, and in the special sciences, figures as the first 



*74 



INTRODUCTION. 



[Book III. 



principle from which all enquiry starts, itself a subject of 
enquiry. 

This important problem philosophy in all ages has kept 
in view and pursued, not indeed with entire success, yet not 
altogether without result ; and assuredly its complete solu- 
tion would be also the completion of philosophy ; for such 
solution could only consist in the establishment of a per- 
fected system of connected truths at once ultimate and 
concrete, from which all the principles which direct the 
researches of the sciences would be derivable, which would 
supply the key to their precise and real significance, and 
define the limits of their validity. It is no such compre- 
hensive undertaking as this, but only a modest portion of 
it, which will form the subject of the concluding chapters 
of this work. Our object is not to enquire into the content 
of the principles in question, but into the grounds upon 
which in a subjective sense their certainty for us reposes ; 
to ask not what is the truth, but what are the marks by 
which we recognise it and distinguish it from error ; or, if 
we are to keep the old terminology, it is our purpose by 
following a method of analysis to obtain clear ideas as to 
the path by which we may hope to arrive at the principles 
of a synthetic development. 

My reason for treating this part of Logic under the head 
of Knowledge, the further discussion upon which we are 
entering will elucidate — an elucidation which the above 
preliminary designation of our undertaking itself certainly 
requires. But in giving it the name of Methodology I 
confess myself to be employing the term to some extent in 
a sense of my own. Every science develops its character- 
istic methods, methods fruitful in their results, which it 
employs in dealing with a given class of problems. But 
Logic regards all such methods as special artifices with 
which it has no concern, but which it belongs to the special 
sciences themselves to furnish. General methods again, 
such as the synthetic and analytic procedure of which I 



Book III.] 



INTRODUCTION. 



175 



have been speaking, do indeed find a mention in Logic ; 
but by formulating them we should only be making a some- 
what barren postulate, until we are clear as to the grounds 
of our belief that the one has actually led us to the dis- 
covery of the truth, and that in the other we have an 
instrument which enables us to develope and exhibit it in 
detail. 

It is this last-named undertaking which I desire to indi- 
cate here by the term ' method/ using it to denote not a 
general type of procedure which has to be applied over and 
over again in a thousand instances, but rather in the sense 
of a definite operation which thought has to go through 
once and for all, and of which the object is to mediate 
between the various sources from which various kinds of 
certainty appear to find their way to us, and to arrive at a 
knowledge of the connexion between them, and of the 
limits of their validity. 



CHAPTER I. 



On Scepticism. 

302. The human mind only becomes aware of the laws 
of its own activity after it has already exerted it in a great 
variety of ways, when it turns back by an act of reflexion 
and comparison upon the various forms which this activity 
has assumed, and makes the rules, which it has been 
following all the while unconsciously, an object of separate 
attention. The question why those laws are binding and 
within what limits their observance carries with it the 
promise of true knowledge, comes still later. It can only 
arise after we have had experience of errors into which we 
appear to have been drawn not by the neglect of those laws, 
but by their observance in dealing with the different subject- 
matters presented to our intelligence. 

If then this has been the case, and if further no success 
has attended our scattered and occasional attempts to re- 
move the difficulties and contradictions which have arisen, 
by giving a better interpretation either to that which seemed 
to us to be truth, or to that which we regarded as the 
immediate deliverance of direct perception, — then arises 
that mood of wide and general doubt which constitutes 
Scepticism. 

As a transient phase of longer or shorter duration this 
sceptical mood has its place in the development of every 
serious mind ; several times in the history of Philosophy it 
has been emphatically insisted on as the normal and 



PRESUPPOSITION OF SCEPTICISM. 177 



necessary condition of the mind, which is called upon at 
the outset of the scientific life to regard all traditional 
knowledge as so much doubtful prejudice, which has to be 
submitted to test and trial. Finally it has established 
itself as a permanent result in the sceptical schools of 
Philosophy, which have believed themselves to have attained 
to the conviction of the impossibility of certain knowledge. 
In this final form in which alone Scepticism pretends to 
have arrived at a definite outcome, we shall not find it so 
entirely free as it flatters itself to be from traditional pre- 
judices. One thing above all however is clear; an un- 
conditional denial of all truth this final outcome of Scepticism 
cannot by any possibility include, inasmuch as not the solu- 
tion of doubt merely but doubt itself is only possible on the 
presupposition of some sort of acknowledged truths. 

Whoever entertains a hope of finding a path out of the 
labyrinth of Scepticism to any form of certain knowledge 
grants this already : for he can only find that path by an 
investigation, and any form of investigation is possible only 
on the assumption at all events of formal principles of judg- 
ment by which one combination of ideas can be dis- 
tinguished as true from a second as false, or from a third 
which is doubtful. And again he who denies that such a 
way out is to be found, in the very act of denial acknow- 
ledges that which he denies. When the old Sophistic 
taught that there was no truth, and that if there were it 
could not be known, and that even if it could be known 
still it could not be communicated, in so doing it contra- 
dicted each of the propositions enunciated. For after all it 
gives out its three propositions for truth, and could not 
therefore deny all truth ; it endeavoured further to prove 
the soundness of its contentions, and was bound therefore 
in its own interest to presuppose the validity of that 
particular form of the apprehension of truth —mediate 
apprehension — the impossibility of which it would have 
been most especially pleasant to point out ; finally it denied 

Logic, Vol. II. N 



i 7 8 



ON SCEPTICISM. 



[Book III. 



the possibility of communicating truth, at the very moment 
when on the strength of its being communicable, it was 
setting itself to convince men of the truth of its own tenets. 

Nor can these contradictions be escaped by avoiding the 
form of positive assertion in the expression of the results of 
the doctrine, and instead of denying the validity of any 
asserted truth simply returning a non liquet to the general 
question equally with all particular ones. Certainly those 
who adopt this course, and we along with them, are at 
liberty to give this answer where the question concerns the 
proof of particular contentions from truths whose validity is 
acknowledged ; but to maintain that the validity of all truths 
whatsoever is doubtful is a proposition which may indeed 
be expressed in words, but the words have no longer any 
real idea to answer to them ; we could not possibly explain 
the meaning of that liquet which we are negativing, if we 
had not in our mind certain conditions under which we 
should be prepared to affirm it, that is to say if we did not 
presuppose some unconditionally valid truth, from which is 
derived our right to doubt whatever cannot be proved to be 
in agreement with it. 

But not only is any sceptical conclusion, in whatever 
form maintained, impossible without this assumption, but 
the very fact of doubt itself is impossible also — impossible 
at least in the only sense in which we are here concerned 
with it. Uncertainty indeed there would be, not sometimes, 
but at least as concerns the future, always and invariably, if 
there were no truth to teach us to distinguish between what 
is necessary and what is not ; but on the other hand we 
should never in that case have occasion to raise the doubt 
whether a given proposition holds as tried by this or that 
standard, inasmuch as it would be a matter of indifference 
whether it did so or not, unless the standard in question be 
recognised as really such, as a veritable criterion, in a word 
as truth. 

However thorough-going then the claims of Scepticism 



Chap. I.] OUR TRUTH AND THE TRUTH. 179 



may be, still it can never get rid, not only of the recogni- 
tion of some absolutely valid truth, but of this presupposi- 
tion also, that the human mind is in possession of certain 
fundamental principles which enable it to affirm at all events 
the impossibility of proving this or that given conception to 
agree with the truth which the sceptic recognises. 

303. To this admission the sceptical mind readily allows 
itself to be driven; it has it will acknowledge a profound 
belief that there is some absolutely valid truth ; and again 
it will grant that necessary laws of thought rule all our 
enquiries and all our doubts : the question which troubles 
it is whether the two — the truth and the laws of thought — 
coincide. Just because we know that there must be truth, 
and therefore that there may be error, how are we to be 
sure that those necessary laws which exist in our mind may 
not belong to the side of error, and everything therefore be 
quite different in itself from that which by the laws of 
thought it necessarily appears to us ? 

It is clear that a scepticism such as this, which is not 
driven to doubt through any special cause residing in the 
nature of its subject-matter, but which simply looks upon 
the possibility of raising a doubt as ground sufficient for 
actually raising it, can never admit of being refuted by 
demonstration. For every argument which can be brought 
into the field against it can only rest upon the self-evidence 
and necessity with which it is thought, and must belong 
therefore to that sphere of necessities of thought as to which 
the old barren question can always be renewed to infinity, 
whether after all things may not be in reality quite other- 
wise than thought makes them. 

This question also has in fact been raised more than once 
in the history of philosophy; at the beginning of the modern 
era by Descartes, who after convincing himself as he thought 
that the soul is furnished with an equipment of innate 
necessary Ideas, presented the question in the following 
vivid form : — might not an evil Demon have so constituted 

N 2 



i8o ON SCEPTICISM. [Book lit 

our nature, that all our thoughts should be necessarily false, 
and yet appear to ourselves clear and necessary truths ? 
And this hypothesis he considered he could only refute by 
pointing to the fact that among these innate Ideas is to be 
found the conception of an absolutely holy and perfect God, 
but, he argued a finite spirit could not have constructed out 
of itself that which is greater than itself, the thought of the 
Infinite; only an actually existing holy God could have 
implanted this in us, and it would contradict the nature of 
this holy God to practise a deception upon us. There 
is one feature in this demonstration which is worthy of at- 
tention ; the underlying thought that in the immediate 
assurance which we feel of the significance of the moral 
Idea lies the security also for the truth of our knowledge : 
but certainly the off-hand way in which the two are thrown 
together here in Descartes' conclusion will convince no one. 
For after all what exception can justly be taken to those re- 
ligious views, which also set out from the belief in a holy 
God, but find it perfectly compatible with the purposes of 
His wisdom in the education of mankind, that He should 
have wholly withheld a large portion of the truth from our 
human knowledge ? And supposing that He had denied to 
us not a portion of truth only but all truth, but in place of 
it had furnished our soul with imaginations which for it are 
necessities of thought, what right should we have had to 
call this withholding of truth and bestowal of error by the 
hard name of a deception, until we had first proved that 
our soul possessed a right to the grant of truth which God 
could not disregard without prejudice to His own holiness, 
and that the apprehension of all existence as it really is, was 
the necessary prerequisite for the fulfilment of those pur- 
poses which we believe that in His holiness He designs to 
accomplish? Such proof Descartes has neither furnished 
nor attempted to furnish ; he abandons himself in the above 
line of thought with all confidence to the guidance of certain 
assumptions which have their limited place in determining 



Chap, I.] THINGS 'IN THEMSELVES?' 181 

questions within the circle of the intercourse of human 
beings with each other, but become mere groundless pre- 
judice when they are applied to that most comprehensive of 
all questions, what is the significance of a necessary law of 
thought, which manifests itself in finite minds; his argu- 
ment would not hinder us in fact from assuming, not indeed 
that a malicious Demon, but that some creative power had 
so fashioned us that all things should actually appear to us 
by a necessity of our thought otherwise than they are. 

Two alternatives are open to us. First we can if we 
please leave any person who is disposed to assent to such a 
hypothesis, to himself, on the ground that we acknowledge 
the impossibility of refuting him, so long as his doubts are 
suggested not by any positive difficulty which renders them 
irresistible, but merely by the possibility of continually re- 
newing them without any positive ground whatever. In the 
presence of this sceptical disposition we should fall back for 
the purposes of science upon a principle from which in the 
ordinary affairs of life our opponent himself cannot escape 
and does not shrink, — faith in reason. We should continue 
to regard a necessity of thought as true until through the 
conclusions which it itself produces it proves itself to be no 
such thing, and compels us to declare it a ' show of being ' 
only, which in such case would be not entirely a vain show 
but an appearance standing in a definable relation to the 
truth with which it can no longer be identified. This atti- 
tude towards the sceptic is that which we find observed in 
life, for through the world's history this groundless scepticism 
has always reappeared from time to time, but as often as it 
has made its appearance men have simply turned their 
backs upon it. 

But in science such a treatment of the question is not al- 
together becoming. The second alternative appears to me 
the more helpful one, to lay bare the essential groundless- 
ness of this curious solicitude, which asks whether after all 
things may not be quite other in themselves than that which 



l82 



ON SCEPTICISM. 



[Book III. 



by the laws of our thought they necessarily appear. What 
after all is the meaning of this addition * in themselves/ or 
this being in itself of something which we oppose to our 
necessary conceptions of the very same something and 
which is supposed to be different from them ? We are here 
in fact, as we now propose to show, in presence of a pre- 
judice springing up from the accumulated effects of experi- 
ence and education, which has crept into the heart of that 
very Scepticism which conceives itself to have got rid of all 
prejudices. 

304. He who begins to reflect upon the foundations and 
the sources of his knowledge finds himself at starting en- 
tangled in all the prejudices which have grown up in him 
unconsciously as his mind has developed, whether arising 
out of his individual experience, or accepted from others. 
For the first attitude of the mind can never be doubt ; it 
begins always with entire confidence in all its perceptions. 
Now no one of these prejudices is more universal than the 
conception of an independent world of things with which 
we habitually contrast our own world of thoughts. Errors 
which meet us within this latter world we regard as trifling 
blemishes easily cured, in comparison with the great and 
dread delusion in which it may be the entire system of the 
world of thought is involved as judged by that other world 
of actual things. 

The doubting question, therefore, whether things may not 
be in fact quite different from what they necessarily appear 
to us, has prima facie an intelligible sense only upon the 
assumption that human knowledge is intended to be a copy 
of a world of things, and in fact that truth regarding the 
possibility of which for man uncertainty is felt, has been 
most commonly defined as the agreement of our ideas with 
the real condition of the things which they profess to copy. 
The ordinary consciousness in practical life never departs 
from this standpoint ; philosophy, on the other hand, in the 
course of its speculations has abandoned it not unfrequently 



Chap. I.] PREJUDICE OF SCEPTICISM. 1 83 

on the strength of knowledge of which it believed itself to 
be already in possession. But a scepticism which in an en- 
quiry into the possibility of knowledge professed to renounce 
all prejudices, was bound above all things not to retain un- 
questioned a definition of the truth of which it was in search, 
founded upon the uncriticised prejudice that there is such 
an external world of things. To dispute that this assump- 
tion is a prejudice is possible only for one who never raises 
a doubt at all, but who feels so complete a satisfaction in 
the direct deliverances of simple perception as to find in it 
at once a convincing evidence of the existence of the external 
world, and an infallible revelation of its nature. But he 
who once entertains a doubt of the truth of any perception, 
and at the same time holds fast as if it were a matter of 
course to the assumption of the existence of the fact, to 
which the perception ought by rights to correspond, can, to 
begin with, only be raising such a doubt at all, on the 
strength of definite convictions as to the nature of the 6 fact' 
in question, convictions which appear to him to be necessi- 
ties of Thought, and which forbid him to take the given 
perception as its true representation. But further, as he 
can no longer regard the thing itself as given him by direct 
perception, it follows that the obligation to retain the belief 
in its existence at all can in its turn only rest upon an innate 
necessity of thought compelling him to supplement and 
complete the manifold world of perception by this thought 
of a world not perceived, in order to bring his ideas in their 
totality into an inward harmony in agreement with the laws 
of his own thought. 

A philosophical review of these questions is necessary, 
not indeed to establish our immediate faith in this world of 
actual things, but to give us scientific justification for hold- 
ing to the assumption of its reality, and on this point the 
systems of Idealism and Realism have arrived at opposite 
results. To bring so comprehensive a problem to an issue 
is not in the least our business here ; on the contrary it is 



184 



ON SCEPTICISM. 



[Book III. 



our purpose to show that as a matter of arrangement the 
question ought not to be imported into these introductory 
discussions on the theory of knowledge. With this view we 
have to consider a single thought in two aspects ; we have 
to remind ourselves in the first place that any decision of 
the question postulates the recognition of the competence 
of thought, secondly we have to show that nothing else but 
the connexion of our ideas with each other can ever be made 
the object of our investigations. 

305. A few words will suffice for the first point. Every 
criticism of the entire apparatus of our faculties of know- 
ledge P, undertaken with a view of enquiring into its agree- 
ment with the nature of things, must presuppose in order to 
its decision a second source of truth Q, which gives us a 
knowledge, free from all alloy of error, of what that nature 
is : for we can only compare known with known, not known 
with unknown. Supposing now that this Q were given us, 
it matters not whether in the form of a comprehensive reve- 
lation imparted originally to our soul, or as a certainty 
coming suddenly upon us as an answer to particular ques- 
tions one by one as they present themselves, in either case 
how are we to compare it with the claims of P which 
requires us to connect our single ideas according to deter- 
minate laws ? 

If P and <2 agree, how could we distinguish the one from 
the other, in order to convince ourselves that it is not only 
our subjective cognition P which is speaking to us, but that 
it has the additional confirmation of the higher objective 
truth <2, evidencing its agreement with the things them- 
selves ? We could not do it at all ; the united utterance of 
the two together would be liable to precisely the same 
doubts and questionings as that of P by itself. If on the 
other hand Q told us something different from P, how 
should we decide between them ? Even supposing that as 
a matter of fact Q gave us truth and P error, how else 
could our faith in the superior credibility of Q be arrived 



Chap. I.] THE CIRCLE OF KNOWLEDGE. 185 



at, except through the greater immediate certainty which 
attended its utterance as compared with that of P? But 
this certainty is inconceivable, except on the condition of Q 
coinciding with that very truth which constitutes the uni- 
versal law of our subjective faculty of cognition P ; what 
contradicts this would, even if it were given us in immediate 
perception, be always a riddle to us, and not a revelation. 
If then Q and P remain opposed, what we arrive at is not 
a refutation of P on the strength of the higher claims of Q, 
but we experience an inner conflict between two utterances 
of that faculty of knowledge which is peculiar to our minds, 
a conflict which either can find no higher court to appeal to, 
and in that case can never be resolved, or must be settled 
by the discovery on the part of that faculty itself of a higher 
point of view within its own province from which one or 
other of the conflicting utterances may be corrected, and 
the apparent contradiction removed. We see then that to 
thought and its necessary laws we are as a matter of fact 
limited in every resort ; the faith which reason entertains 
that truth whatever it may be is discoverable by thought, is 
the unavoidable postulate of all enquiry; what that truth is 
can be discovered only by the reflective operation of thought, 
continually trying and testing its single results by the stan- 
dard of the universal laws of its activity. 

306. It is in vain to shrink from acknowledging the circle 
which is here involved, for that there is no escape from it 
everyone after all must see. It is also superfluous, because 
there can never come a moment in our experience — and 
this is the second point we have to urge — in which the sup- 
posed mischief which our vague suspicions apprehend could 
possibly become known to us. All we know of the external 
world depends upon the ideas of it which are within us ; it 
is so far entirely indifferent whether with Idealism we deny 
the existence of that world, and regard our ideas of it as 
alone reality, or whether we maintain with Realism the 
existence of things outside us which act upon our minds. 



i86 



ON SCEPTICISM. 



[Book ill. 



On the latter hypothesis as little as the former do the things 
themselves pass into our knowledge ; they only awaken in 
us ideas, which are not Things. It is then this varied world 
of ideas within us, it matters not where they may have come 
from, which forms the sole material directly given to us, 
from which alone our knowledge can start. In them, and 
in the course which they follow as they change and connect 
themselves, we endeavour to discover a regular and orderly 
arrangement, guided in our search by those universal prin- 
ciples of our thought which determine for us what we are to 
account as order and truth, and what as involving a contra- 
diction or a problem to the reason. 

Every discovery of such a law, regulating universally and 
without exception any two determinate ideas 1 and iMn 
their conjunction with each other in our minds, is the attain- 
ment of a fragment of that which we call knowledge of Fact 2 . 
If we fail in our effort to discover such a constant connexion 
between the two, then we have a problem before us, which 
we always set to work to solve in the same way. First we 
endeavour to find some universal relation between B and 
what is contained in a third idea M, and between Fand 
what is contained in a fourth idea IV, and then to show that 
by reason of a variable relation which obtains between M 
and N, that between B and F cannot be expressed in the 
form of a simple law such as we were looking for, but only 
through a law of a different kind which takes account of M 
and TV as well. 

If finally we are in doubt whether a relation which we 
have discovered to exist between two ideas B and F in our 
minds, corresponds to the reality of things, this can only 
mean that we doubt whether whenever B and F reappear as 
ideas in our consciousness the relation between their con- 
tents which we have collected from only a limited number 
of instances, will continue to obtain universally and without 
exception. But if the question be once more repeated : is 
1 [' Vorstellungen.'] 2 [Sache.] 



Chap. I.] 



WHAT A MISTAKE IMPLIES. 



187 



a relation between B and F as established for consciousness 
even by invariable experience also true in itself, such a 
question is only intelligible at all on one supposition, namely 
if the relation existing as a matter of fact in consciousness 
does not accord with the universal postulates of thought, — 
those postulates which thought constrains us to make in the 
case of any relation between any two objects whatever, and 
therefore also of those which we are endeavouring to con- 
ceive as obtaining between real existences independent of 
ourselves. It is not this assumed external world of the Real 
which comes in here between our ideas as the standard by 
which their truth is to be measured ; the standard is 
always the conception of which we cannot get rid, of what 
such a world must be if it does exist, is always that is to 
say a thought in our own minds ; this it is by which we 
measure the truth of other thoughts, whether they contain 
the evidence in themselves, or are such as to require elu- 
cidation from without. 

307. It is perhaps superfluous, but it may not be without 
its use, to repeat this simple argument, starting from the 
opposite side, and to ask what it is that must happen if we 
are to discover a supposed piece of knowledge Z to be a 
delusion. Suppose we knew from our own observation that 
between two ideas B and F frequently recurring in our ex- 
perience the relation Z does not uniformly obtain, but on 
the contrary varies according to the varying relations in 
which B is found conjoined with Af, and Fwith JV. Sup- 
pose on the other hand that another human being lived 
within a sphere of experience where those conditions exclu- 
sively obtain under which the relation Z between our B and 
F does become a necessity. It will follow that he will never 
have occasion to doubt the universality of that relation Z, 
nor will his faith in it prejudice the coherence of the rest of 
the world of his ideas, provided only that Zdoes not conflict 
with the universal laws of his thought. Unquestionably the 
assumption that Z is an absolute relation between B and F 



i88 



ON SCEPTICISM. 



[Book III. 



independently of further conditions, may make it much 
more difficult for him to find a simple law for the relations 
between other constituent elements of his experience, as C 
and E, which he would at once discover if he was aware of 
the dependence of Z upon conditions which do also in fact 
determine the relation between C and E. But so long as 
he does not extend his faith in Z beyond the objects con- 
tained in the world of his ideas, he will still be in a position 
to systematise the objects connected in that world, however 
awkwardly he may have to express their connexion. We 
indeed who possess the experience which he lacks are aware 
of his error, but we can only convince him of it by taking 
him out of his more limited circle of experience and trans- 
planting him into a wider. Then when he himself finds 
fresh conjunctions of ideas arising in his mind distinct from 
those which he formerly experienced, he will allow that he 
has been in error ; still all he will have to concede will be 
that he was mistaken in supposing the relation Z between B 
and F to obtain universally; that relation still holds true 
when the conditions are added upon which, though un- 
awares to him, its validity all the while depended. 

And now how will it be if we place human reason as such 
in the position of this unfavourably situated observer, and 
imagine it confined to a mode of mental representation, 
coherent indeed in itself, but not corresponding to the 
real relations which obtain in a world of things external 
to it? How is the standing delusion, in which in that 
case we are all involved, to become known to us, or how 
will our knowledge suffer supposing it to continue ? Setting 
aside for the moment the instruction which an angel from 
heaven might impart to us, what we find is this : it is 
certainly not the things themselves which are here making 
their way all of a sudden into the midst of our thoughts, 
and laying bare their falsity; even if the world of things 
running its independent course were to enter some day 
upon a new arrangement which diametrically contradicted 



Chap. I.] THE TRUTH OF OUR DELUSION. 189 

the conceptions which we had previously formed of it, 
such contradiction could only come within our observation 
through the new influences awakening in us a set of ideas 
which we find no longer to observe those laws of com- 
bination which we had hitherto assumed to be their laws. 
Then we have fallen into one of those errors of the under- 
standing to which we of course allow that we are liable ; 
we have wrongly interpreted this variable world of ideas, 
that world which is the sole material that lies open to our 
intelligence ; we recognise now that we have learnt a new 
lesson, and that the proposition Z does not possess the 
universal validity with which we had credited it ; but we 
have learnt also that it does continue to be valid still 
when the conditions of that validity with which we have 
now become acquainted are reckoned in. And now the 
universal validity of Z being erroneous, so also its limited 
validity is true, and we come to see that inasmuch as error 
can only be observable by us through an inner contra- 
diction in our world of ideas, it follows that the recognition 
of truth itself consists only in the discovery of laws of 
connexion which this ideal world is destined always to 
observe, to however infinite a distance we may imagine 
its varying course to be prolonged. Undoubtedly the dis- 
covery of these laws is an undertaking which must remain 
incomplete ; we are not in possession of the whole truth, 
we are in search of it ; still so often as we correct a previous 
belief Z at the instance of fresh experiences in our world 
of thought, we have not indeed reached as yet the full 
truth, but we have removed the errors which without such 
correction would have lasted on. 

308. This argument, unless I am much deceived, will 
satisfy no one. We are left after all, it will be urged, even 
if all inner contradictions are removed, walled in within 
the all-embracing delusions of those ideas which have 
grown up into a solid mass within us, and never see the 
truth in itself, but only as it necessarily appears to us. 



190 



ON SCEPTICISM, 



[Book III. 



Now then let us call in our angel from heaven, who 
beholds from his purer atmosphere things as they are. 
What a shock we fancy it would be to us if all at once 
he withdrew the veil from our eyes, and we saw how 
entirely different things really are from what we had 
imagined them. 

And indeed we should experience a very agreeable shock 
if in that moment it were revealed to us how our old ideas, 
with all their old meaning, by the introduction of certain 
simple links in the chain hitherto concealed from us, became 
at once perfectly intelligible, with no gap or contradiction 
remaining, and intelligible by the light of the old laws 
which had all along directed the movement of our thought. 
But also on this condition only. If it were an entirely new 
world which rose upon our view, bearing no resemblance 
and no relation to that in which we had lived before, we 
simply should not perceive that everything was different 
from what we thought ; for what that meant to us was that 
everything we thought was different from what we thought 
it ; the wholly new spectacle admitting, as it would, of no 
comparison with the former one, could on this ground at 
all events give us no shock at all, pleasurable or otherwise ; 
it could not so much as occasion us surprise, except through 
a sense of contrast, that is to say by being brought into 
relation with our previous world of illusion. But again 
we who now see ought to be the same persons who before 
were blind. If that moment of revelation had at the same 
time transformed the laws of our thought, and altered the 
conditions under which hitherto we had distinguished truth 
and error, we should indeed, if our newly discovered world 
completely answered to these new conditions of truth, have 
no occasion to doubt about any particular fact in it ; but 
what could preserve us from the grand fundamental doubt, 
whether this new world of ideas with all its self-consistency 
may not in its turn distort the true nature of reality, and 
things be once more quite different in themselves from 



Chap. I.J 



TRUTH IS NOT THE THING. 



that which in this new world they appear to be ? Do you 
hope to exclude these doubts on the ground that on our 
own assumption it is the truth of things themselves, and 
nothing else, which makes up the content of our new per- 
ceptions? But to exclude the possibility of doubt, the 
fact that our representation of things is the true one would 
not of itself suffice. We must also possess means to arrive 
at a certain knowledge that it is the true one. Now such 
means we do possess in regard to particular parts of our ex- 
perience ; we can measure their truth by asking, are they 
as judged by the universal laws of our thought in harmony 
with the rest of that same experience ? But it is impossible 
to test the truth of the entire world of our ideas as such by 
comparing it with a reality which so long as it is not an 
object of knowledge is for us non-existent, and if once it 
becomes so must be subject to the same doubts and un- 
certainties to which all ideas simply as such are liable. 

And finally the supposed case is in itself impossible and 
absurd. What can be the meaning of saying that this 
higher intuition, perception, cognition, gives the thing in 
itself, as it really is? We may exalt the intelligence of 
more perfect beings above our own as high as we please ; 
but so long as we desire to attach any rational meaning to 
it, it must always fall under some category of knowledge or 
direct perception, or cognition, that is to say it will never be 
the thing itself but only an aggregate of ideas about the 
thing. Nothing is simpler than to convince ourselves that 
every apprehending intelligence can only see things as they 
look to it when it perceives them, not as they look when no 
one perceives them ; he who demands a knowledge which 
should be more than a perfectly connected and consistent 
system of ideas about the thing, a knowledge which should 
actually exhaust the thing itself, is no longer asking for 
knowledge at all, but for something entirely unintelligible. 
One cannot even say that he is desiring not to know but to 
be the things themselves ; for in fact he would not even so 



192 



ON SCEPTICISM. 



[Book III. 



reach his goal. Could he arrive at being in some way or 
another that very metal in itself, the knowledge of which in 
the way of ideas does not content him ; well, he would be 
metal it is true, but he would be further off than ever from 
apprehending himself as the metal which he had become. 
Or supposing that a higher power gave him back his intelli- 
gence while he still remained metal, even then in his new 
character of intelligent metal he would still only apprehend 
himself in such wise as he would be represented to himself 
in his own ideas, not as he would be apart from such repre- 
sentation. 

309. In dealing with these fundamental questions I 
ought not to be blamed for the lengthened discussion 
which I have permitted myself. It is true the outcome 
is small. We have convinced ourselves that this changing 
world of our ideas is the sole material given us to work 
upon ; that truth and the knowledge of truth consist only 
in the laws of interconnexion which are found to obtain 
universally within a given set of ideas, and are confirmed 
as often as those ideas recur in our consciousness ; that as 
the thoughts which lead us towards this order of truths 
make way, the antithesis between our ideas and the objects 
to which we conceive them to be directed, itself a part of 
that same world of ideas, necessarily arises ; that the ques- 
tion as to the truth of this antithesis, and the value which 
according as we answer it will belong to our ideas, is a 
question of metaphysics which has no business to be mixed 
up with an introduction to the theory of knowledge such as 
the present ; that in regard to this or that among our 
thoughts we may doubt as to the possibility of bringing it 
into harmony with the rest of the content of our conscious- 
ness, and that such doubts resting on definite grounds are 
compatible with the endeavour gradually to remove them ; 
that on the other hand a scepticism which indulges the 
apprehension that everything may be in reality quite diffe- 
rent from what it necessarily appears, sets out with a self- 



Chap. I.] REASONINGS OF SCEPTICISM. 193 



contradiction, because it silently takes for granted the 
possibility of an apprehension which does not apprehend 
things but is itself things, and then goes on to question 
whether this impossible perfection is allotted to our intelli- 
gence. Finally we see that if we set aside this inadmissible 
relation of the world of ideas to a foreign world of objects, 
there still remains a further line of enquiry open to us, — the 
endeavour to discover within the world of ideas itself what 
are the fixed points, the primary certainties, starting from 
which we may be enabled to bring the rest of the shifting 
multitude of its ideas into something like orderly connexion. 
I shall find and shall avail myself of various opportunities 
hereafter for elucidating this point of view ; I go on at pre- 
sent to glance at the different methods of procedure which 
the sceptical philosophy has followed, and which have been 
pursued in the various departments of enquiry to which 
they have been applied, upon the whole with greater com- 
pleteness in antiquity than in modern times, when many 
of its questions are no longer able to excite an active 
interest. 

310. Sextus Empiricus has left us a collectanea of the 
tenets of Scepticism down to his own time. The Sceptic 
does not any more than other men deny the sensuous per- 
ceptions, the feelings of pleasure and pain, which we experi- 
ence. They force themselves irresistibly upon him, and are 
independent of his opinion. On the other hand everything 
that is contrasted with these phenomena, as a noumenon, or 
as a thought, which itself not given in the phenomenon, 
seeks to bring the content of the perceptions into some 
inner connexion, — all this is made open to doubt, and any 
statement we venture from this point of view may be met 
with equal propriety by another which contradicts it. No- 
thing therefore remains for the wise man but to refrain from 
either affirming or denying either the one proposition or the 
other, and to find in this suspension of judgment that peace 
of mind which so long as he considers it his duty to decide 

Logic, Vol. IL O 



i 9 4 



ON SCEPTICISM. 



[Book III. 



between two conflicting hypotheses, he must necessarily 
seek in vain. 

But when Scepticism, not content with representing an 
abstention from any affirmation as the condition of mind 
actually found in its adherents, undertakes to prove it on 
logical grounds to be the only legitimate attitude of the 
mind, it becomes at the very outset false to itself, pre- 
supposing as it does at all events at this point the truth of 
those logical laws of thought by which alone it can establish 
the cogency of its own reasonings. And not only so but in 
its efforts to expose the impossibility of dogmatic statements 
it is compelled to assume a variety of dogmas which can 
never be directly given in phenomenal experience, but can 
only be derived from them by those very processes of 
reasoning whose legitimacy is contested. The ten rpoiroi or 
logical grounds of doubt, which Sextus begins by rehearsing, 
all come to this, that sensations by themselves cannot dis- 
cover to us what is the nature of the object which excites 
them. The first rporros calls attention to the different or- 
ganisations of different animals ; when it goes on further to 
the proposition that by reason of this any object must appear 
different to the senses of one animal from what it does to 
those of another, it appeals to the Dogma that unlikes can- 
not be affected alike by likes. Nothing short of this argu- 
ment would have justified his conclusion, for as we cannot 
place ourselves inside an animal's consciousness, this sup- 
posed difference between the sensation of one animal and 
that of another is a conclusion given by reasoning, which 
can never be established by immediate perception. More 
than this, the argument affirms too much ; there is nothing 
to prove that visible differences in bodily organisation are 
an invariable indication of corresponding differences of 
feeling ; no one will easily believe that a cat, by reason of 
its elliptically shaped pupil, must necessarily perceive the 
world of space differently to a man with his circular one. 

The second rponos applies the same argument within the 



Chap. I.] SCEPTICAL rpojroi ABOUT OBJECTS. 



195 



circle of human beings. They too in their turn are variously 
organised; if then, the rpoVoy argues, it were proposed — 
though we have no grounds for so doing — to give the human 
sensations a preference over the animal, and to regard these 
as true and adequate to the thing itself, we are again de- 
feated by reason of the individual differences which exist 
between man and man. So that all we can say is, that to 
one man the thing appears in one way, to his neighbour in 
another way; how it is in itself remains unsettled. 

The two next rponoi lead to the same result ; the third 
appeals to the differences among the senses themselves ; to 
the eye honey is yellow, to the tongue sweet ; it may be that 
there are other forms of sensation, lacking to us, to which it 
appears something different again : what it is in itself must 
therefore be relegated to uncertainty, as we have no reason 
for accounting the deliverance of one sense truer than that 
of another. Even supposing however that we keep to a 
single sense, the fourth rponos points out that here too there 
are variations of feeling, according to age and state of 
health, according as we are hungry or satisfied, asleep or 
awake, so that still we can only say how a thing appears to 
our sense under each of these varying conditions, but not 
how it would appear to a subject which was experiencing 
none of them. 

These four rpoVot were concerned with the nature of the 
subject which frames judgments ; the four which follow re- 
late to the objects. The fifth reminds us that distance and 
position alter the appearance of one and the same thing : 
the sixth points out that no object produces in us the im- 
pression of itself unmixed with those of others ; the seventh 
that the composition of various elements in single objects 
causes qualities to appear in them which are entirely wanting 
in the simple elements themselves, and effaces others which 
belonged to them ; so that we can never do more than state 
how each one appears in its several combinations with other 
things, nor what it is like in itself and by itself apart from 



196 



ON SCEPTICISM. 



[Book III. 



the various phases which by reason of those relations it 
passes through. It is impossible to read the examples to 
these last rpoVot without a feeling of astonishment that the 
scepticism of antiquity should have seen in them throughout 
only impediments to scientific knowledge. In modern 
science they have become one and all starting-points of 
enquiry. Modern science has not been content with rais- 
ing a general lamentation over the changeableness of phe- 
nomena under changing conditions \ it has questioned 
experience ; it has enquired what are the special con- 
nexions which obtain between any one of these conditions 
and this or that particular change in the phenomenon, and 
it has in this way arrived at a knowledge of the general 
laws which govern this endlessly changing play of events. 
We have not indeed learnt what a thing is like in itself when 
it stands wholly apart from all the conditions of its mani- 
festation to intelligence ; but that the problem so presented 
is absurd the ancient scepticism was itself aware, as we find 
it expressed in the eighth rpoVo? : Everything stands in rela- 
tions of one kind or another, if not to other things, yet 
always at least if it is to become an object of apprehension, 
to the subject apprehending it ; what it is like in itself, 
apart from all relations, remains therefore beyond our power 
to say. 

The last two rpoW are of less interest for us : the ninth 
reminds us that our estimate of the magnitude and the 
value of things is conditioned by their rarity or frequency, 
by custom and by contrast ; the tenth appeals to the diver- 
sity of national manners and morals as an evidence that 
here too we can dnly say what appears good and bad to 
one person or another, not what is good and bad in itself. 

311. The further development of the Pyrrhonian vitotv- 
7Tft)cr€tff of Sextus, from whose first book the doctrines I have 
cited are taken, I here pass over. It will here be evident 
that so far the Scepticism we have been considering does 
not deny the reality of truth, for it is the impossibility of 



Chap. I.] 



DOES SCEPTICISM JUDGE ? 



197 



attaining to truth which it laments over, and one can only 
make that an object of quest in the reality of which one be- 
lieves. Nor does it doubt that conformity to the laws of 
thought is the necessary condition of any thought being 
true. It is incessantly enumerating, in disjunctions alleged 
to be complete, sets of cases which are inferred on the basis 
of these laws to be possible and to exclude one another; 
and it is by this same logic of thought that it undertakes to 
bring us to acknowledge the necessity of withholding judg- 
ment altogether. It is true that this procedure has to be 
subsequently corrected. The sceptical argument is at pains 
to include itself in the uncertainty to which, by one of those 
very affirmations which it seeks to get rid of, it condemns 
the whole of our pretended knowledge. The forms of argu- 
ment which are employed for this purpose are many and 
curious. If the Sceptic arrives at his negations by a process 
of demonstration, he is not, it is said, in so doing, here any 
more than elsewhere, laying down any positive doctrine ; he 
is simply stating that to him, here and now, at this particular 
moment of his life, and in the particular state of mind in 
which he happens to be, the opinion which he has an- 
nounced appears to be the true one. He does not guarantee 
its continuing so to appear even to himself at every future 
moment ; if he is driven to acknowledge some one else's 
argument to be convincing, he can always answer, the truth 
Z which this man teaches, has up to this moment been 
unrecognised, yet all the same if it is truth it has been so 
always and been always valid ; and where is our security 
that some third person may not hereafter discover and de- 
monstrate a new truth to upset Z in its turn, which at the 
present moment, though it already holds good, is neither 
recognised nor capable of being either apprehended or 
proved ? 

These questions are independent of the relation of our 
knowledge to an object outside itself; they concern the 
ground of certainty generally, and our right to the con- 



198 



ON SCEPTICISM. 



[Book life 



fidence we repose in the truth of any thought in our minds ; 
in this view we reserve them for consideration later. But 
apart from this the arguments of Sextus involve at once a 
prejudice and a fallacy; the prejudice of the existence of 
that World in itself with which knowledge was contrasted, a 
prejudice which may be just or the contrary — that cannot be 
decided here — the fallacy that the conception of a knowledge 
which apprehends things not as they are known but as they 
are, means anything intelligible at all, as to the possession 
or non-possession of which it is possible to raise a con- 
troversy ; whereas the truth is that upon this at least thought 
is perfectly clear and at one with itself, that knowledge 
under whatever form can never be things in themselves but 
only represent them. 

312. There will be a disposition to express this contention 
in the form that we only know phenomena, and not the 
essence of things in themselves, and so stated to recognise 
it as the primary truth of every theory of cognition. I avoid 
that particular form of statement because it still contains a 
prejudice which I should wish to see abandoned. The 
actual assumption indeed of the existence of this world of 
things which is given by the categorical form of the propo- 
sition might be avoided by transforming it into the hypo- 
thetical : If things exist knowledge apprehends only their 
appearance, not their essence. But even then the propo- 
sition plainly carries the idea of a thwarted purpose. That 
'only' implies that our knowledge which was intended by 
rights for the apprehension of the higher, the essence of 
things, has to be content with the lower, the phenomenon. 
Such a valuation is once more a prejudice, it may be legiti- 
mate, it may be not legitimate, as the further progress of 
Science may decide which we are not here in a position to 
anticipate. But we can see at once that it is an arbitrary 
proceeding to place knowledge in the position of a means 
which is not adequate to its supposed end of apprehending 
things as they are. And we may at once pronounce an 



Chap. I.] 



1 ONLY' PHENOMENA. 



199 



opposite point of view to be conceivable, which should 
regard things as mere means to produce in us in all its 
details the spectacle of the ideal world. If this were so we 
should not indeed know things as they are, but we should 
not therefore fail of any end or aim ] in the appearances 
which things present to us would reside then that element 
of higher dignity and value which we sought to indicate by 
the name of essence ; and in the discovery of the purport, 
the connexion and the laws which govern this inner world 
of phenomena, the knowledge of truth would lie not indeed 
exclusively but pre-eminently, and at least as truly as in that 
which we are now so painfully anxious to arrive at, the 
apprehension of that which must always remain outside our 
own and every other intelligence, the system of means 
through which the series of phenomena is called into exist- 
ence within us. But to continue this discussion further 
would be to overstep the limits of my undertaking. I 
repeat once more what I desire the reader to understand 
this to be; let us leave entirely out of the question the 
opposition between our world of ideas and a world of 
things \ let us look upon the former alone as the material 
we have to deal with ; and let us endeavour to ascertain 
where within this world the primary fixed points of certainty 
are to be found, and how it may be possible to communi- 
cate a like certainty through the medium of these to other 
ideas which do not in themselves equally possess it. By 
following certain circuitous paths which will be found to be 
no deviation from our proper route, we may perhaps arrive 
at clearness on this subject. 



CHAPTER II. 

The World of Ideas. 

313. The problem which we have set before us is one 
which ancient philosophy long ago declared again and again 
to be insoluble. That all is in flux was the familiar doctrine 
of Heraclitus, a doctrine however of which it is difficult to 
determine the precise significance. That it was understood 
in the half pathetic sense of a lamentation over the rapidity 
of change appears in the heightened form subsequently 
given to his saying that it is impossible to cross the same 
river twice — ' it is impossible,' it was added, 4 even once.' 
But against the testimony of observation to the transitori- 
ness of things the most ordinary experience might have set 
counter examples of duration through incalculable periods 
of time ; philosophical reflexion could only have universal- 
ised the former set of experiences into the doctrine cited by 
establishing in opposition to superficial appearances, that 
the latter also do but veil a slow process of change to which 
in fact they are always subject. We do not know how far 
this actually took place and whether these speculations 
passed over without notice the circumstance that the differ- 
ences in the speed of one set of changes and another at 
once introduce into the play of phenomena a contrast be- 
tween the relatively fixed and the more transitory which 
might be turned to fruitful account. Once more, that no- 
thing can wholly withstand agencies of change operating 
from without, that everything therefore must be susceptible of 



HER ACUTUS VIEW OF CHANGE. 201 



change, is a conviction too easily derived from the experi- 
ence of every-day life to have needed a philosophy to 
discover it. But it remains doubtful how far Heraclitus 
passed beyond this, and taught that there are changes in all 
things springing from causes in their own nature and not 
merely occasioned by outside influences, and whether he 
taught this simply as a fact of experience, or whether he 
held continual movement to be the condition of the possi- 
bility of all natural existence, and that stable equilibrium 
and permanence were impossible. 

There is much to lend probability to a view which should 
credit him with this more advanced conception, but the 
question can as little be certainly decided as the more im- 
portant one what precisely is to be understood by the 'all' 
to which he ascribed this ceaseless mutability. The expres- 
sion included beyond question the things of sense ; in fact 
the very starting-point of the doctrine could have been 
found nowhere else but in the changing combinations of 
sensible qualities and relations. But did it include at the 
same time the content of the ideas by means of which we 
think this world of sense ? Was it intended that not only 
all that is real but all that can be thought as well is subject 
to this eternal flux ? I doubt if Heraclitus held this latter 
opinion ; the universal instability of all determinations of 
thought would of course render all enquiry and all affirma- 
tion impossible. We may however assume from the lively 
picture which Plato draws in the Theaetetus of the later ac- 
tivity of the school, that they at all events had no hesitation 
in giving this extension to their master's doctrine. 

At this point it is taken up by the Sophists. I do not 
mean that section which under the leadership of Protagoras 
acknowledged only the subjective validity of every percep- 
tion for the person who experiences it, I mean those who, 
disciplined in the Eleatic dialectic, set themselves to demon- 
strate that every conception signifies at once what it does 
mean and what it does not mean. This contention was 



202 



THE WORLD OF IDEAS. 



[Book III. 



met, principally in the field of Ethics, where it produced its 
most pernicious effects, by the sound instinct and sense of 
truth of Socrates, who called attention to the fact that the 
conceptions of good and bad, just and unjust, are fixed and 
unchanging, and cannot be determined now one way and 
now another at the pleasure of individuals, but that they 
have to be accepted as permanent and self-identical con- 
ceptions to which everyone has simply to subordinate his 
own ideas on these subjects. Plato followed, at one with 
these aims of his master, but impelled by more many-sided 
motives, and expanded the convictions received from So- 
crates into his own doctrine of Ideas, a first and most 
characteristic attempt to turn to account the truth which 
belongs to the world of our ideas in itself, without regard to 
its agreement with an assumed reality of things outside its 
borders. The philosophical efforts of antiquity have the 
attraction of exhibiting in full detail the movements, the 
struggles, and the errors of thought, into which every indi- 
vidual still falls in the course of his development, and which 
notwithstanding the culture of our own day has no longer 
the patience to follow up and investigate. I shall permit 
myself to enter therefore into a review of this doctrine of 
Plato, approaching it at various points which seem pertinent 
to our present enquiry. 

314. The Platonic expression Idea 1 is usually rendered 
Universal conception 2 , and the rendering is so far correct 
that there are Ideas, according to Plato, of everything which 
can be thought in a universal form, apart from the par- 
ticular perceptions in which it is presented. At the same 
time it is only for the purposes of a later set of conceptions 
which we shall meet with presently, that it becomes impor- 
tant to be able to think of the ideally apprehended content 3 
as something common to many individual contents, that is 

1 [' Idee.' Where the term ' Idea ' represents ' Idee ' and not 
' Vorstellung ' it is printed with a capital ' I.'] 

3 I' AllgemeinbegrinV] 3 [< Inhalt.'] 



Chap. II.] 'IDEAS' AS 'OBJECTIVE CONTENTS. 1 203 



as a universal. What is essential here at the outset is not 
so much that it can be separated from different particular 
instances which contain it, as that it has been distinguished 
as a content with a meaning of its own which we present to 
ourselves, from a mere affection which we experience. In 
the latter sense it might have been involved by the Hera- 
clitic or pseudo-Heraclitic doctrine in its ceaseless flux of 
events, of which each one only is in the moment in which 
it occurs, and no one has an abiding habitation or signifi- 
cance in the world, because there is no reason why having 
once occurred it need ever recur again in identically the 
same form. The former conception on the contrary turns 
the mere affection of our sensibility into an independent 
objective 1 content whose significance once is its significance 
once for all, and whose relations to other contents have an 
eternal and self-identical validity even if neither it nor they 
should ever be repeated in actual perception. 

I have had occasion to explain my meaning here in an 
earlier part of this work (§ 3). Perception shows us the 
things of sense undergoing changes in their qualities. But 
while black becomes white and sweet sour, it is not black- 
ness itself which passes into whiteness, nor does sweetness 
become sourness ; what happens is that these several quali- 
ties, each remaining eternally identical with itself, succeed 
each other in the thing, and the conceptions through which 
we think the things have themselves no part in the muta- 
bility which we attribute on account of their changes to the 
things of which the qualities are the predicates — and even 
he who attempted to deny this would be affirming it against 
his will, for he could not represent sweetness as passing into 
sourness, without separating the one property from the 
other, and determining the first for his own thought in an 
idea which will always mean something different from the 
second into which it is supposed to have changed. It is a 
very simple and unpretending, but yet a very important 

L 1 v. § 3.] 



204 



THE WORLD OF IDEAS, 



[Book III. 



thought to which Plato here gives expression for the first 
time. The continual change which goes on in the external 
world may affect us like a restless whirling eddy, bewildering 
our intelligence, yet it is not without a pervading truth. 
Whatever mutability the things may display, that which 
they are at each moment they are by a transient partici- 
pation in conceptions which are not transient but for ever 
identical and constant, and which taken together constitute 
an unchangeable system of thought, and form the first ade- 
quate and solid beginnings of a permanent knowledge. 

For it was one of the conclusions at which we arrived 
before 1 , that to the making of this earliest immediate stock 
of knowledge there contribute not merely the separate unity 
of each conception in itself, nor again simply the fact of a 
mere uniform contrast between this and all other concep- 
tions, but also those graduated relations of resemblance and 
affinity in which different conceptions stand to each other. 
If the white becomes black and the sweet sour they do not 
merely become different in the abstract, but pass over from 
the domain of the one conception in which they participated 
before into that of another which is separated from the first 
by a fixed and determinate degree of contrast, a contrast 
stronger for example than that which obtains between 
white and yellow, and altogether incommensurable with 
that absolute gulf of separation which exists between white 
and sour. 

315. I refer to these simple examples once more in order 
to make it clear how a knowledge may be possible the truth 
of which is wholly independent of the question of Scepti- 
cism as to its agreement with a world of things outside it. 
If the current of the outer world had brought before us 
only once in a transient appearance the perception of two 
colours or two sounds, our thought would immediately 
separate them from the moment of time at which they 
appeared, and fix them and their affinities and their con- 
1 [Cp. §§ 13-16.] 



Chap. II.] 



KNOWLEDGE OF 'IDEAS? 



205 



trasts as an abiding object of inner contemplation, no 
matter whether they were ever presented to us again in 
actual experience or not. Again supposing we could never 
learn how these ideas are able to appear as predicates in 
things, and in what that which we have called the partici- 
pation of things in them exactly consists, a question would 
indeed be left unanswered which might in the course of 
our reflexions prove important, but still the certain know- 
ledge would remain to us undisturbed that the series of 
colours and the scale of musical tones themselves are each 
a connected whole with fixed laws, and that in regard to 
the relation of the members to each other, eternally valid 
true propositions are vitally opposed to eternally invalid 
false ones. And finally the question whether after all 
colours in themselves and tones in themselves are not diffe- 
rent from what they appear to us, is one which no one will 
care to raise again. Or rather we do meet with it again in 
the confused notion that sounds are in fact merely vibrations 
of the air, colours merely quiverings of the ether, and it is 
only to us that they appear in the form of the subjective 
feelings which we know. It is unnecessary to enlarge over 
again on the consideration that these feelings do not cease 
to be real, and are not got rid of and banished out of exist- 
ence as intruders, because we have discovered certain ex- 
ternal causes not resembling them, which are the occasions 
of their making their appearance to us. Even if these 
vibrations of external media appeared to differently con- 
stituted beings in the form of modes of sensation entirely 
unknown to us, still the colouring and tones which we see 
and hear, would constitute for us, when once we have ex- 
perienced them, a secure treasure of knowledge with a 
validity and an orderly connexion of its own. The feelings 
of such other beings would remain unknown to us and ours 
to them, but this would only mean that we have not all 
truths for our portion, but that what we do possess we 
possess as truths in virtue of the identity of every such 



20Ö 



THE WORLD OF IDEAS. 



[Book IIL 



content of perception with itself, and of the constancy of 
identical relations which obtain between different contents. 
Thus we readily understand the significance of Plato's en- 
deavour to bind together the predicates which are found in 
the things of the eternal world in continual change, into a 
determinate and articulated whole, and how he saw in this 
world of Ideas the true beginnings of certain knowledge ; 
for the external relations which subsist between different 
Ideas, and through which some are capable of association 
with each other and others exclude each other, form at all 
events the limits within which what is to be possible in 
experience falls ; the further question what is real in it, and 
how things manage to have Ideas for their predicates, ap- 
peared to Plato not to be the primary question, and was for 
the time reserved. 

316. There is one wide-reaching difficulty connected with 
the first-named aspect of this question. How precisely are 
we to conceive colours when they are not seen, or tones and 
their differences when the former are not heard and the 
latter not apprehended by comparison ? Are we to say that 
they are nothing or that they do not exist, or are we still to 
attribute to them some predicate which we can hardly de- 
fine, some kind of being or reality? We shall not be dis- 
posed at first to consider them to be nothing at all ; for as 
long as we fix them in our thoughts, as at present in search- 
ing for an answer to this very question, every tone and every 
colour is a determinate content distinguishable from every 
other, and so a something and not a nothing. Still this 
decision becomes doubtful when we consider the answer 
which we feel ourselves compelled to give to the second 
part of the question. In regard to things we do imagine 
ourselves, dimly enough, to know wherein their being con- 
sists even when they are objects for no intelligence, but 
exist purely for themselves ; but what is meant by a tone 
when it is heard by no ear and when even the silent idea of 
its sound is not called up by any mind, we can no more 



Chap. IL] 



EXISTENCE AND REALITY. 



207 



say than what a pain is when no one is hurt by it. But 
how can that which is not either in itself or in our con- 
sciousness, be any longer anything at all or be distinguished 
from anything else ? Still this conclusion again we hesitate 
to affirm. There is clearly in our first conclusion, speaking 
quite generally, a certain element of affirmation, which is 
not entirely to be cancelled by the denial contained in the 
second. Perhaps it may appear to us a way out of the 
difficulty to turn the categorical form of our judgment into 
a hypothetical; two sounds which are neither heard nor 
imagined are not indeed actually anything, and stand in no 
actual relations, but they will always be something and the 
one will be different from the other, and stand in a definite 
relation of contrast to it, if they are heard or imagined. 
Yet even this does not at once satisfy us, for in order even 
to imagine how the notes a and b can be subject to this 
varied fortune of being presented to imagination at one 
time and not so presented at another, and then how it 
happens that when they are presented in experience the 
relation Z is necessarily thought along with them, whereas 
whenever certain other sounds are presented, they are no 
less necessarily accompanied by a different relation Z 1 , — in 
order to imagine this we are constrained to ascribe to them 
existence and definite existence, at a time when according 
to this view they did not in fact exist at all ; for so alone 
can we explain their subsequent existence and the definite 
form which their relations then assumed. 

I will not pursue these refinements further, but will con- 
clude with the following remarks. We have undoubtedly a 
conception of affirmation or ' position ' in an extremely 
general sense, which meets us in various fields of enquiry, 
and for which languages, dealing as they do in their early 
stages with highly complex and concrete notions, and not 
with the simplest elements of thought, have commonly no 
abstract term which expresses it with the requisite purity. 
But it would not be wise to invent a technical term to 



20S 



THE WORLD OF IDEAS. 



[Book III. 



represent it, the meaning of which would always be doubt- 
ful, because it could never come naturally to the lips or to 
the thoughts of any one; the very term 1 position' which is 
frequently used for it suggests by its etymological form the 
entirely alien sense of an act, or operation of establishing 1 , 
to the execution of which that state of affirmation which we 
wish to express then seems to owe its being. It is best 
however to keep to ordinary speech, and select a word 
which can be shown to express in common usage, approxi- 
mately at all events and unmistakeably, the thought with 
which we are concerned. We may express it in our own 
language by the term Reality 2 . For 3 we call a thing Real 4 
which is, in contradistinction to another which is not ; an 
event Real which occurs or has occurred, in contradis- 
tinction to that which does not occur ; a relation Real 
which obtains, as opposed to one which does not obtain ; 
lastly we call a proposition Really true which holds or is 
valid as opposed to one of which the validity is still doubt- 
ful. This use of language is intelligible; it shows that 
when we call anything Real, we mean always to affirm it, 
though in different senses according to the different forms 
which it assumes, but one or other of which it must neces- 
sarily assume, and of which no one is reducible to or con- 
tained in the other. For we never can get an Event out of 
simple Being, the reality which belongs to Things, namely 
Being or Existence, never belongs to Events — they do not 
exist but occur ; again a Proposition neither exists like things 
nor occurs like events ; that its meaning even obtains like a 
relation, can only be said if the things exist of which it pre- 
dicates a relation ; in itself, apart from all applications 
which may be made of it, the reality of a proposition 
means that it holds or is valid and that its opposite does 
not hold. 

1 [< Setzung.'] 2 [< Wirklichkeit/] 

3 [Cp. ' Metaphysic,' p. I, and for « Objectivität ' contrasted with 
different forms of ' Wirklichkeit ' see above, § 3.] 

4 ['Wirklich.'] 



Chap. IL] 



REALITY AND VALIDITY, 



209 



Now misunderstandings must always arise, when under 
the persuasion that the object which we are considering 
must have some sort of reality or affirmation proper to it, 
we endeavour to attribute to it, not that kind of reality 
which is appropriate to it, but a different kind which is 
alien to it. Then arises the conflict just noticed between 
the conviction on the one hand that we are right in as- 
scribing to it some sort of reality, and on the other that 
the particular form of reality to which our misconception 
has brought us is inadmissible. 

Now Ideas, in so far as they are present in our minds, 
possess reality in the sense of an Event, — they occur in us : 
for as utterances of an activity of presentation they are never 
a Being at rest but a continual Becoming ; their content 
on the other hand, so far as we regard it in abstraction from 
the mental activity which we direct to it, can no longer be 
said to occur, though neither again does it exist as things 
exist ; we can only say that it possesses Validity. 

And finally we must not ask what in its turn is meant 
by Validity, with any idea that the meaning which the 
word conveys clearly to us can be deduced from some 
different conception; as if, for example, it were possible 
to find certain conditions by the operation of which either 
the Being which belongs to things could be so modified 
and attenuated, or the momentary act of Becoming or 
occurring, in which the transient reality of ideas regarded 
as excitations of our consciousness consists, could receive 
such fixity and independent existence, as that both the 
one and the other in different ways might pass into this 
conception of Validity, which at once excludes the sub- 
stance of the valid assertion from the reality of actual being 
and implies its independence of human thought. As little 
as we can say how it happens that anything is or occurs, 
so little can we explain how it comes about that a truth 
has Validity; the latter conception has to be regarded 
as much as the former as ultimate and underivable, a con- 

Logic, Vol. IL p 



2IO 



THE WORLD OF IDEAS. 



[Book III. 



ception of which everyone may know what he means by 
it, but which cannot be constructed out of any constituent 
elements which do not already contain it. 

317. From this point of view some light I think is 
thrown on a surprising statement which is handed down 
to us in the history of Philosophy. Plato, we are told, 
ascribed to the Ideas of which he had achieved the con- 
ception an existence apart from things, and yet, as 
these same critics tell us, of like kind with the existence 
of things. It is strange how peacefully the traditional 
admiration of the profundity of Plato acquiesces in the 
ascription to him of so absurd an opinion ; we should have 
to abandon our admiration of him if this really was the 
doctrine that he taught, and not rather a serious mis- 
understanding to which in a quite intelligible and pardon- 
able way it has laid itself open. The expression of philoso- 
phical ideas is dependent upon the capabilities of each 
language, and it is hardly possible, in giving utterance to 
our meaning, to avoid using words which language has 
coined to express a merely cognate thought which is not 
our real meaning at all. And this is pre-eminently the 
case when a new field is being opened out, and the 
necessity of distinguishing the precise meaning intended 
from the ordinary meaning of the word is as yet little 
felt. This is I think the explanation of the misunder- 
standing in question. The truth which Plato intended to 
teach is no other than that which we have just been 
expounding, that is to say, the validity of truths as such, 
apart from the question whether they can be established 
in relation to any object in the external world, as its mode 
of being, or not ; the eternally self-identical significance of 
Ideas, which always are what they are, no matter whether 
or no there are things which by participation in them make 
them manifest in this external world, or whether there are 
spirits which by thinking them, give them the reality of a 
mental event. But the Greek language then as afterwards, 



Chap. IL] IDEAS WHY ETERNAL AND SEP AR A TE. 2 1 1 

was wanting in an expression for this conception of Validity 
as a form of Reality not including Being or Existence ; and 
this very expression Being came, often indeed quite harm- 
lessly, but in this instance with momentous consequences, 
to fill the place. 

Every possible content of thought, regarded as an indi- 
vidual unity, distinct and separate from others, all that class 
of things for which the language of the School philosophy 
in later times invented the not inappropriate name of 
Res rationis 1 was to the Greek a Being (pv or ovo-la) • and 
if the distinction between a really valid truth and a pre- 
tended truth came in question the former was distinguished 
as optcos ov. The language of ancient Greece never found 
any term to express the reality of simple Validity as dis- 
tinguished from the reality of Being, and this constant 
confusion has prejudiced the clearness of the Platonic 
phraseology. 

318. We may easily see that everything Plato says of the 
Ideas presents itself when understood in the manner so 
explained as natural and necessary, and that the various 
devices to which he resorts in setting forth their nature have 
this purpose and no other, to exhaust the conception for 
which no adequate term could be found, by the help of a 
variety of expressions limiting and supplementing each 
other. Eternal, without beginning, and imperishable (aldia, 
dyewrjTa, avcokeSpa) the Ideas could not but be named in the 
presence of the flux of Heraclitus, which seemed in danger 
of sweeping them away along with the sense-world in its 
stream. The reality of Being indeed they have or have not, 
according as transient things of sense are clothed with them 
or not ; but that reality which consists in Validity, which is 
a reality all their own, remains untouched by all this change. 
Their independence of time, when brought into comparison 
with that which comes and goes in time, would hardly be 
otherwise expressed than by this predicate of eternity which 
1 [' Gedankending.'] 
P 2 



212 



THE WORLD OF IDEAS. 



[Book III. 



at once partakes of time and denies its power, just in the 
same way as we should most easily recognise that which has 
no validity and could have no validity in itself by the fact 
of its never occurring at any moment of time. 

Again, we understand the ideas being called separable or 
separate from things (x<»pis tw ovtcov), first because the image 
(ddos) of their content can be still called up to memory after 
the things which originally occasioned its appearance in us 
have vanished from real existence, and next, because the 
content is taken to include what can be apprehended in a 
universal form, and remains the same in different external 
manifestations, so as to be independent of the mode in 
which it is realised to sense in any particular instance. 

But it was not Plato's intention to represent the ideas as 
independent merely of things while still depending for their 
special mode of reality upon the mind which thinks them. 
Reality of Existence it is true they enjoy only in the 
moment in which they become, in the character of objects 
or creations of an act of presentation now actually occurring, 
members of this changing world of Being and Becoming ; 
but on the other hand we all feel certain in the moment in 
which we think any truth, that we have not created it for 
the first time but merely recognised it ; it was valid before 
we thought about it and will continue so without regard to 
any existence of whatever kind, of things or of us, whether 
or not it ever finds manifestation in the reality of Existence, 
or a place as an object of knowledge in the reality of a 
Thought. This is what we all believe with regard to truth 
when we set out to search for it, and it may be lament over 
its inaccessibility at least to any form of human knowledge ; 
the truth which is never apprehended by us is valid no whit 
less than that small fraction of it which finds its way into 
our intelligence. 

The independent validity of the Ideas Plato emphasises 
again in a somewhat different form, in answer to the doctrine 
of Protagoras, rescuing them in their character of being in 



Chap. IL] 



THE vorjTos tottos. 



213 



themselves that which they are (avra Kaff avra opto) from the 
relativity in which the famous dictum of that Sophist was in 
danger of involving them. Even granting that his doctrine 
has its truth so long as it is confined to the impressions of 
sense, and that viewed in this relation Plato's opposition to 
it rests upon a misunderstanding, granting that is to say that 
my sensation is as true for me as yours which differs from it 
is for you, Plato would still be right in insisting that for 
neither of us could the sensation be possible at all, unless 
that which we felt in the sensation whatever it be, red or 
blue, sweet or bitter, had a definite and constant significance 
of its own, as a member of a world of Ideas. This world 
of Ideas is the permanent and inexhaustible treasure-house 
from which the things of the external world draw all the 
diverse and shifting attributes they wear, and the mind the 
varying series of its experiences ; and a sensation or idea 
whose content has no fixed and determinate place, no fixed 
relations of affinity or difference in the universal world of 
thought, but stands in complete isolation, bare of all relations 
to anything in that world, the possession of a single individual 
mind alone, is in fact an impossibility. 

While Plato by thus describing the Ideas, takes security 
for their independent validity, he has at the same time 
abundantly provided against the confusion of the validity 
thus implied with that wholly distinct reality of Existence 
which could only be ascribed to a durable thing. When he 
places the home of the Ideas in a super-celestial world, a 
world of pure intelligence (votjtos, vnepovpavLos tokos), when 
again more than this he expressly describes them as having 
no local habitation, such language makes it abundantly clear 
to any one who understands the mind of Greek Antiquity, 
that they do not belong to what we call the real world. To 
the Greek that which is not in Space is not at all, and when 
Plato relegates the Ideas to a home which is not in space, 
he is not trying to hypostasize that which we call their mere 
validity into any kind of real existence, but on the contrary 



2I 4 



THE WORLD OF IDEAS. [Book III. 



he is plainly seeking to guard altogether against any such 
attempt being made. Nor is it any objection that the Ideas 
are called unities (eVdSey, iiovddts), for there is no occasion to 
interpret these titles from an atomistic standpoint, whether 
in the sense of material indivisibility or of a self-identity 
resembling that of a self-conscious subject. For in fact 
what constitutes the meaning of an Idea, and of a complex 
no less than of a simple Idea, is that it manifests itself as a 
unity, unifying the elements which cohere in it and rejecting 
that which is alien to it. Nevertheless although these 
various expressions point one and all to the fact that Plato 
never asserted the existence of the Ideas but only their 
eternal validity, he had still no better answer to make to the 
question, what then are they, than to bring them again 
under the general denomination of oixria. Then the door 
was opened to the misunderstanding which has since widely 
spread, though no one has ever been able to say what the 
nature of that existence, into which he is accused of having 
hypostasized his ideas, precisely is. 

319. There are two objections which may be taken to the 
view here maintained. First, the use which Plato makes of 
the Ideas to explain the course of the world, in which they 
assert their influence not merely as valid truths but as 
operating forces — this is a point to which I shall come 
later; and in the second place, the attitude of Aristotle. 
For it is really the very definite language of Aristotle which 
has established the doctrine of the reality of the ideas as a 
dogma of Plato, whereas Plato's own statements are in no 
way inconsistent with the other interpretation which we 
have preferred. It seems incredible that the most acute of 
Plato's disciples, informed by personal intercourse with the 
master, should have misunderstood him in a point of such 
serious moment as this. At the same time we are justified 
by the nature of his polemic not against particular statements 
of Plato but against the doctrine of Ideas altogether, as well 
as by many details in his criticisms, in assuming that his 



Chap. II.] 



ARISTOTLE'S ovvla. 



215 



attack is in part directed against certain misunderstandings 
of the Platonic doctrine which had gained hold in the 
Academy at an early period. For he could not well have 
challenged Plato himself to show where the Ideas are, when 
Plato had said in plain terms that they were nowhere. He 
could not have directed against Plato the criticism that there 
must logically be Ideas of products of art, for one passage at 
least is to be found in the Republic which is entirely in agree- 
ment with that criticism, and how far Plato was from having 
overlooked the difficulty there involved, is evidenced by the 
opening of the Parmenides. Finally as to Aristotle's ob- 
jections to the Ideas that they are superfluous, being mere 
copies of individual objects, and the assumption from which 
his elaborate analysis frequently starts, that there are as 
many examples of every Idea as there are instances of its 
application in reality, these are criticisms which do not really 
apply to the doctrines of Plato himself. That every Idea is 
what it is once for all, that what we are to understand by it 
is not an individual thing but a universal comprehending 
many things, and that all its manifestations are only copies 
of this one essential reality, is the doctrine which he 
never abandons, whatever obscurity may still attach to that 
operation on the part of the individual things, described as 
imitation or participation, by which they provide the one 
Idea with a countless number of realisations in the world of 
actual existence. 

The discussion therefore which fills the Xllth (XHIth) 
book of Aristotle's Metaphysics and of which the purport is 
to exhibit the absurdity of attributing to the Idea a reality 
identical with the reality of actually existing things, I cannot 
regard as a refutation of the pure Platonic doctrine, and the 
less so inasmuch as at the end Aristotle himself equally fails 
to find a decisive and unambiguous expression for that 
more appropriate form of reality which he desires, in contra- 
distinction to this, to ascribe to them. To him the only 
genuine ovala is the individual thing, and there we must 



2l6 



THE WORLD OF IDEAS. 



[Book III. 



certainly agree with him; to the individual thing alone 
belongs the reality of Existence ; still for Aristotle as much 
as for Plato the object of knowledge is always the universal; 
not only in the sense that we are incapable of exhausting 
the meaning of the individual thing, but that so far as we 
investigate it in its nature and its workings with any prospect 
of a result, we invariably proceed according to universal 
principles. But Aristotle is entirely at one with his pre- 
decessors, that that which is not, or has no reality in any 
sense, cannot be an object of knowledge either, and so in 
regard to the universal we cannot say that it simply is not, 
but that in a sense it is and in a sense it is not. 

I do not propose to enter into Aristotle's further treatment 
of this question in detail. I must however remark that by 
placing the universal and the Idea within the Individual 
things and not outside them he does not explain the possi- 
bility of knowledge ; for the mere fact of the presence of the 
Idea in one individual does not entitle us to transfer all the 
consequences which flow from it to a second individual in 
which it happens also to be found ; it can only justify us in 
concluding from the doings of one real thing to those of 
another, if it includes within itself a number of characteris- 
tics so related that the appearance of any one necessarily 
implies the presence of the rest. Such considerations would 
at once conduct Aristotle back to the admission that the 
Idea is certainly in a sense x^pi* T <» v optchv; but in what 
sense it is so was impossible for him to define, since he no 
more possessed than his master did a technical equivalent 
for our term validity ; and thus eventually the universal 
conception or Idea came to be for him also an oiaia, not 

indeed a true Or 7rpü)Trj ova-la but Still a devrepa ovala. 

320. It may appear to us a strange spectacle to see two 
of the greatest philosophers of antiquity struggling with 
imperfect success to arrive at clearness upon so simple a 
distinction as that which we have been considering. But 
such a view would do both of them injustice. The appre- 



Chap. IL] THINGS AND LAWS OF NATURE. 217 



hension of the simplest relations of thought is not the 
simplest act of the faculty of thought, and the whole long 
history of philosophy teaches how ready we all are at any 
moment to be guilty of a degree of obscurity in the applica- 
tion of ideas which if reduced to its simplest terms would 
appear to us incredible. Whenever men have believed 
themselves to have discovered a principle which appears to 
represent the universal element in the constitution and 
development of the real world, they invariably go on to 
exalt it into the position of an independent reality and to 
represent it as a pure form of being, in comparison with 
which the individual things retire into a position of subor- 
dinate and even unreal existence. I need not even refer to 
the latest phase of German philosophy which aspired to set 
on the throne of the Platonic Ideas the one absolute Idea, 
for the same tendency is apparent enough in spheres of 
thought outside the circle of philosophy. How often do we 
hear in our own day of eternal and unchangeable laws of 
nature to which all phenomena and their changes are 
subjected; laws which would indeed cease to manifest 
themselves if there were no longer any things for them to 
control, but which would even then themselves continue in 
their eternal validity and would revive with their old effective 
power the moment a new object presented itself from any 
quarter for them to apply to ; nay there is not even wanting 
on occasion, the enthronement of these laws above all 
existing realities in that very super-celestial habitation which 
with Plato is the home of the Ideas. Nevertheless those 
who hold this language would indignantly repel the imputa- 
tion of ascribing to those laws an existence whether as things 
or as persons outside the things which are governed by 
them, and Plato may resist with equal justice a similar 
misinterpretation of his doctrines. 

Finally it must be added that we ourselves, in drawing a 
distinction between the reality which belongs to the Ideas 
and laws and that which belongs to things, and calling the 



2l8 



THE WORLD OF IDEAS. [Book III. 



one Being or Existence 1 and the other Validity 2 , have so 
far merely discovered, thanks to the resources of our 
language, a convenient expression which may keep us on 
our guard against interchanging the two notions. The fact 
which the term validity expresses has lost none of that 
strangeness which has led to its being confounded, as we 
have seen, with existence. It is merely that we have been 
so long accustomed to it ; we use our thought as we do any 
other natural faculty without troubling ourselves about it, 
and take it as a matter of course that the content of manifold 
perceptions and phenomena does invariably adapt itself to 
general conceptions and can be read by us in the light of 
general laws, in such wise that the consequences which 
those laws lead us to predict are found to coincide with the 
actual phenomenal order which supervenes. But that this 
should be the case, that there should be universal laws, 
which have not themselves existence like things and which 
nevertheless rule the operation of things, — remains for a 
mind which realises its meaning, a profoundly mysterious 
fact which might well inspire rapture and wonder in its 
discoverer ; and that he should have made the discovery 
will always remain a great philosophical achievement of 
Plato, whatever the problems it may have left still un- 
solved. 

321. One of these problems is that of the exact nature 
of the relation of things to the ideas which Plato describes 
by the terms participation or imitation. I do not propose 
at present to discuss this question at large ; but there is one 
defect in the doctrine of the Ideas which a criticism of 
Aristotle's — in itself not well-founded — may suggest to us. 
Among the reasons which led him to regard the Ideas as 
both superfluous and useless, he especially emphasises the 
fact that they supply no beginning of motion. However 
true this objection may be in itself, the fact that they do not 
perform this task proves little against the doctrine of the 
1 [< Sein.'] 2 [< Geltung.'] 



Chap. IL] IDEAS AND BEGINNING OF MOTION 219 

Ideas ; the real objection is that they do not, as we shall 
see, adequately perform the task for which Plato intended 
them. As concerns Aristotle's criticism let us turn to the 
sciences of our own day. What shall we say of our Laws 
of Nature ? Do they contain in themselves a beginning of 
motion? On the contrary, they all presuppose a series 
of data which they cannot themselves establish, but from 
which, once given, the necessary connexion one with another 
of the phenomena which ensue is deducible. No natural 
law ordains that the different bodies in our planetary system 
should move, or that their course should be directed towards 
one and not another quarter of the heavens, or that the 
acceleration which they impose on each other by the force 
of attraction should have the particular amount which it has 
and not a different one. But is the whole system of 
mechanical truths useless and mere empty babble (Ktvokoyfiv) 
because it leaves these first beginnings of motion to be 
explained from some other source, and starting from the 
fact of motion as it actually finds it, is satisfied with explain- 
ing its different phases in their necessary connexion with 
each other ? There may be obscurity enough — though 
after all not more than in our own mode of representing 
the matter — in Plato's relegation of the primary motive 
impulses upon which the succession of phenomena depends, 
to that dim world of v\t) which represents to him the 
material which is given for the Ideas to be applied to. But 
for all that to see in the world of Ideas the patterns to 
which all that is, ^/anything is, must conform, was a thought 
of which the importance is unfairly ignored by Aristotle. 
For he was himself on a later occasion to have recourse to 
that very same thought, for the explanation of individual 
phenomena : he too found himself unable to allow the 
cause of motion which communicates the actualising im- 
pulse also to control its issue ; this had been decided from 
all eternity by those universal laws, which in their turn take 
no part in the communication of the impulse. 



220 



THE WORLD OF IDEAS. 



[Book III. 



On the other hand it must undoubtedly be admitted to 
be a deficiency in the Platonic doctrine that this, which 
was its actual undertaking, it only half accomplishes. An 
account of the necessary connexion of two contents of 
thought must always assume the logical form of a judgment; 
it cannot be expressed in the form of a mere notion which 
does not in itself contain a proposition at all. Thus we 
have always employed laws, that is to say propositions, 
which express a relation between different elements, as 
examples to explain the meaning of Validity in contradis- 
tinction to Existence. The term cannot be transferred to 
single concepts without some degree of obscurity : we can 
only say of concepts that they mean something, and they 
mean something because certain propositions are valid of 
them, as for example the proposition that the content of 
any given concept is identical with itself and stands in un- 
changeable relations of affinity or contrast to others. Now 
Plato apprehended the elements of the world of thought 
which he discovered almost exclusively under the form of 
the isolated concept or the Idea. We need not look beyond 
the general impression which his Dialogues leave with us to 
be aware how rarely by comparison we meet with general 
propositions ; they are by no means entirely absent, on the 
contrary they are made on occasions the subject of impor- 
tant disquisitions, but that it is propositions as such and 
nothing else which must necessarily form the most essential 
constituents of the ideal world, is a truth which never 
forced itself upon Plato's mind. His peculiar point of view 
is not without modern parallels. Kant himself in his search 
for the a priori forms which were to give the unity of an 
inner coherence to the empirical content of our perceptions, 
made the mistake at starting of developing them in the 
form of single concepts, the Categories, and that in spite of 
the fact that he derived them from the forms of the judg- 
ment itself. And now having got them, as he thought, in 
his Categories, it became the more evident that there was 



Chap. IL] IDEAS ARE NOT PRINCIPLES. 221 



nothing to be made of them, and thereupon followed the 
attempt to derive judgments out of them again, and so he 
arrived at the ' Principles of the Understanding' which it 
was now possible to apply as major premises to the minor 
premises furnished by experience. It seems therefore that 
this disposition to bring into the inadequate form of a single 
concept truths which can only be adequately expressed 
through the proposition, is natural to the imagination at all 
times, and is not peculiar to the plastic mind of ancient 
Greece. It may however be remarked in passing how 
dangerous a tendency it is, leading the mind as it does away 
from the full concrete reality which is the true aim of its 
enquiries to a barren playing with empty ideas which have 
become separated from their natural foundations. 

Thus we find our present requirements hardly at all 
satisfied in Plato, and even the need of satisfying them not 
clearly or adequately recognised. It is true the abstract 
thought that the Ideas are not only a multitude of indi- 
viduals but that they make up all together an organic and 
articulated whole — this thought is the soul of all his teach- 
ing, and he describes with enthusiasm the delight which he 
finds in the dialectical exercise of resolving the complex 
structure of the Ideal world into its elements, following the 
natural joinings, and then putting them together again ; 
even the different degrees of agreement or of contrast 
between individual ideas and the possible modes of com- 
bining them are mentioned as subjects worthy of investiga- 
tion. But in the examples which he gives of the applica- 
tion of his method, the art of Dialectic ends almost 
invariably in a mere classification of Ideas, by which we are 
shown the place which belongs to any one Idea in a system 
of division in virtue of the elements which it combines, but 
which furnishes us with no single proposition, adds no jot 
to our knowledge concerning the nature of any one of the 
Ideas which could not have been arrived at equally without 
this circuitous route of classification. If we want to know 



222 



THE WORLD OF IDEAS, 



what can be said or cannot be said of any Idea we have 
still to learn it, after the classification as much as before, 
from other sources. The joinings and articulations of 
truth which Plato's sole aim was not to mutilate he ought 
to have investigated with a firmer hand ; instead of making 
a systematic collection of the flora of the Ideas, he ought to 
have turned his thoughts to the general physiological condi- 
tions which in each single plant bind limb to limb according 
to a law of growth. Or, dropping the figure, the existence 
of a world of Ideas possessing a definite meaning and an 
unchangeable validity being once clearly and emphatically 
established, the next task was to investigate the universal 
laws which govern its structure, through which alone, in an 
Ideal world as elsewhere, the individual elements can be 
bound together into a whole. Thus the question to be 
dealt with at this point was what are those first principles 
of our knowledge under which the manifold world of Ideas 
has itself to be arranged. This is the more precisely 
defined form which the systematic enquiry into Truth and 
the source of Truth now assumes for us. 



CHAPTER III. 

The a priori and the Empirical Methods. 

322. When we feel in doubt about any particular point 
of belief within the sphere of our knowledge we endeavour 
to clear it up by analysing the conditions which have led us 
to entertain the belief ; we expect to learn from the history 
of its origin whether it is true, or if it is false how it must 
have grown up. And whenever in the history of philosophy 
the question has arisen as to the capacity of the human 
mind for the attainment of truth in general, mankind have 
thought that the same path would lead there also to the 
goal. It has been supposed that the claims of our ideas 
and our judgments to the name of truths could be decided 
by considering the process by which they have been formed. 
This belief, which is worth considering inasmuch as it lies 
to a great extent at the root of certain tendencies of philoso- 
phical enquiry even in our own day, leads me to quit for 
the moment the subject upon which I have entered. It is 
necessary to attempt to point out, that this method of 
criticising our Ideas by tracing their genesis does not 
present the advantages as applied to the subject of human 
knowledge generally which it does undoubtedly possess in 
the case of particular beliefs or ideas. 

For the two cases are not alike. If we desire to test the 
accuracy of any particular opinion, we have a basis for our 
decision in other truths of which we are in acknowledged 
possession, on the one hand general principles with which 



224 A PRIORI AND EMPIRICAL METHODS. [Book III, 

all other propositions if they are to have validity for us must 
be in agreement, on the other hand established facts, which 
must not be contradicted by those other facts which are 
either affirmed or assumed by the view under question ; 
finally we have certain laws of thought by which, given 
certain valid premises, logical conclusions derived from 
them are distinguished from illogical. Throughout we start 
from some truth which operates upon the mixture of our 
thoughts which is submitted to the test like a fermenting 
matter, assimilating that which is akin to it, and rejecting 
that which is alien. Such a standard given us to start with 
and itself independent of the subject-matter of enquiry, is 
! wanting when we turn to the larger problem ; to test the 
truth of human knowledge in general is impossible, without 
assuming as the basis for our decision the very principles 
which are on their trial. 

This logical circle according to which our knowledge has 
itself to determine the limits of its own authority, we have 
already seen to be unavoidable; but we increase our 
difficulties, if, instead of regarding those principles them- 
selves as the one element of certainty in our knowledge, 
from the vantage-ground of which we may go on to take 
possession of the rest of its domain, we explicitly attribute 
this certainty not to those principles themselves but to a 
particular unanalysed application of them, viz. to our 
supposed insight into the origin of our knowledge. The 
theory is that the mode in which knowledge originates 
is to decide its claims to truth, that truth moreover, as is 
supposed by this view, having regard to a reality which is 
foreign to and transcends knowledge. But ' if this is our 
aim we cannot move a single step without making certain 
more definite assumptions ; first as to the position of the 
knowing subject as regards those objects of its knowledge, 
next as to the nature of that relation, between it and those 
objects, by which the process of knowledge is carried on. 
For it is only by understanding these circumstances that we 



Chap. III.] 



ORIGIN OF KNOWIEDGE. 



225 



can learn to estimate the dangers which stand in the way of 
the formation of true conceptions. 

The pretence therefore of setting to w T ork to ascertain the 
process by which knowledge comes to us, by a simple act 
of observation, discarding all prejudice and eschewing all 
admixture of principles whose validity can be called in 
question, is in fact a groundless illusion. Every attempt to 
carry out such an undertaking is necessarily full of meta- 
physical assumptions, but assumptions disconnected and 
uncriticised, because they are merely taken up at the 
moment as they happen to be wanted to clear up a 
difficulty. The circle is inevitable, so w r e had better per- 
petrate it with our eyes open ; the first thing we have to do^ 
is to endeavour to establish what meaning it is possible for 
us to attach to knowledge in its widest sense, and what sort 
of relation we can conceive to subsist between the subject 
which knows and the object of its knowledge, consistently 
with those yet more general notions which determine the 
mode in which we have to conceive the operation of any- 
thing whatever upon anything else. What we have to do is 
to obtain the last-mentioned conception, which amounts to 
a metaphysical doctrine, and to treat the relation of subject 
and object as subordinate to it ; we are not to begin by 
setting up some chance theory more or less probable as to 
that one relation, and then to use this as a test of the 
capacity of the human mind for apprehending truth at all. 
I say nothing of the question how far it is really in our 
power even to establish the facts regarding the gradual 
development of our world of thought ; certainly the process 
of that development cannot be directly observed, for every 
observer has left it long ago behind him. And even though 
in many cases the developed consciousness may still retain 
the recollection of the road by which it has come to its 
present set of ideas, it will be admitted on the other hand 
that in many other cases these pretended observations of 
the development of our ideas are merely somewhat fanciful 

Logic, Vol. II. Q 



2 26 A PRIORI AND EMPIRICAL METHODS. [Book nr. 

theories of the mode in which we think we may conceive it 
to have taken place. 

323. If we follow the attempts which have been made to 
arrive in the first instance at some fact beyond the reach of 
doubt, from which we may proceed with security to test the 
origin and the truth of human knowledge, we are met at the 
outset of modern philosophy by the maxim of Descartes, 
* cogito ergo sum,' the one certain truth which the doubt of 
all received opinions seemed to him to leave standing. 
This proposition has been frequently taken as a point of 
departure, and it has always approved itself, from as far 
back as Augustine, in whose writings we first find it, for a 
truth as unquestionable as it is absolutely barren. Not the 
smallest step towards the establishment of any theory of 
knowledge whatever has it been possible to take from this 
proposition by itself, without calling in other and wholly 
independent principles to help. The very criterion which 
follows next in order, that all ideas are true which are 
equally clear and evident with this, Descartes himself did 
not venture to derive from that primary principle, without 
securing himself by the roundabout argument, alluded to in 
a previous chapter, against the objection that we may be all 
the while deluded by entirely false ideas possessing an equal 
degree of evidence with the true. 

In point of fact it is easy to see that from this beginning 
we never can get to anything further. If we take the 
proposition in its negative sense, that is to say that nothing 
is certain for us except the fact of our own thought alone, 
and there is no such certainty in regard to the real existence 
of an externa] world, then I recall an observation already 
made : even if such external world be really existent, still it 
is only an ideal picture of it and not that world itself which 
can be present in us : the fact therefore that nothing 
possesses immediate certainty for us excepting our own world 
of thought, can never settle the question whether it alone 
exists, or whether there is a world of existence outside it 



Chap. III.] 



CO GITO ERGO SUM. 



227 



to which it enters into relation. And even if the idea of 
this external world could be proved to be a necessary 
product of our thinking activity which we are compelled to 
form through the organisation of our mind and the laws of 
interconnexion to which our thoughts necessarily conform, 
if that is to say we could deduce from the fact of the cogito 
that our assumption of an external world of existence must 
necessarily have a subjective origin in the laws of our own 
minds : even then the truth of the assumption would be 
neither proved nor disproved ; for even if that external 
world does really exist, it would be impossible for us to 
arrive at the idea of it unless the nature of our mind and 
the workings of our thoughts were such as to render it 
indispensable for the avoidance of contradiction within the 
world of thought itself. 

On the other hand if we turn our attention to the affirma- 
tive aspect of the proposition, we find that it is not formu- 
lated in a way adapted to its purpose. It is no longer the 
expression of an immediate fact but of an abstraction. I do 
not complain of Descartes for keeping to the first person of 
the verbs ' cogito ' and ' sum, 7 for obscure as the idea of the 
' ego 9 may be which they contain, and provocative as it is 
of further enquiry, it does unquestionably belong to the 
original form of this simplest of all experiences, and a 
theory which seeks to supplant the 'cogito' by the ' cogitare 9 
and the 'sum 9 by the 'esse 9 as the primary and most certain 
fact of experience, has no claim whatever to the credit of 
resting on a basis free from all presupposition and prejudice, 
which it is its ambition to share with the exact methods of 
the natural sciences. There never meets us as the simplest 
of facts an idea which merely exists and which no one has ; 
we never meet with a consciousness which presents itself 
simply as consciousness and not as the consciousness of an 
'ego 9 which in it is conscious to itself either of itself or of 
something else. Science may attempt afterwards to separate 
by one means or another the occurrences of thought and 

Q 2 



228 A PRIORI AND EMPIRICAL ME TIIODS. [Book in. 

knowledge from this their constant condition of reference 
to a subject whose nature remains impenetrable : but they 
are originally given and their certainty along with them only 
in the form ' cogito* 4 1 think,' not in that of the infinitive 
' cogitare? But while Descartes is entirely correct in em- 
ploying the personal form of the verb, it must be acknow- 
ledged that its significance was overlooked by him, and the 
interpretations which it received at the hands of Kant we 
cannot enter into here. 

We may add that Descartes' principle was expressed in 
an unserviceably abstract form, emphasising as it does in 
the various mental states which carry with them this imme- 
diate certainty of personal experience, exclusively their 
universal quality — that is to say it emphasises exclusively 
the fact of cogitation or consciousness in the widest sense, 
which is an element entering equally into very various 
mental states, sensations and ideas, emotions and the will, 
distinguishing them all alike from that which we suppose 
ourselves to conceive as the condition of a being without a 
self and without a soul. No doubt this element of con- 
sciousness enters into every one of our mental states which 
we observe, but what can be the use of noticing this 
common quality alone to the ^exclusion of those concrete 
elements apart from which it cannot really exist or become 
an object of direct observation at all ? 

The really fruitful starting-point of enquiry would have 
been, not the fact that the ' cogito' is found in every form 
which consciousness can assume, but the question, what are 
the forms in which it is found ? Not the bare fact that we 
are conscious or think teaches us the truth we know; it is 
what we think, the matter or content of our cogitation, 
which supplies not only the original datum from which we 
start, but the sole source from which that which we ought 
to think or that which we cannot but think can be derived. 
Descartes himself points out that even the Sceptic in his 
doubt or in his denial of all knowledge, by that very act 



Chap. III.] 



INNATE IDEAS. 



229 



confirms the fact of cogitation, and just because it is asso- 
ciated indifferently with all true knowledge, and with every 
act of doubt, and with every kind of error, it cannot possibly 
serve to distinguish the true from the false. 

324. Thus a fresh starting-point for the enquiry into 
human knowledge was unavoidable. It was given by the 
belief in the truth of innate Ideas. We must not allow the 
expression 6 innate Ideas,' which has introduced a long con- 
troversy into the history of the theory of knowledge, to 
excite prejudices in our minds which a little care and 
consideration may certainly allay. Even the ancients in 
speaking of that quod a natura nobis insitum est, and all 
philosophers who have used the like expressions, were 
certainly very far from assuming that a truth, in itself 
foreign to the mind, was stamped in upon it at some par- 
ticular moment when its life was beginning, and became 
thenceforth a permanent object of its conscious thought. 
What they meant was no more than this — the mind is of its 
own nature so constituted, that under certain operative 
conditions it necessarily develops certain habitual modes of 
combining its ideas. These constitute, to begin with, a 
method which the mind follows unawares, but finally as it 
comes to reflect upon innumerable acts of thought per- 
formed in accordance with them, the rules of its procedure 
hitherto unconsciously followed become themselves the 
objects of its conscious reflexion. These Ideas were called 
innate from the impression that it was not sufficient to 
represent the mind in which they were supposed to grow 
up as merely possessing a certain formal character, or 
general capacity for ideas, in such a way that given the 
same conditions the same set of Ideas would necessarily 
grow up in every being so endowed ; it was held essential 
that every mind should have its determinate natural capa- 
city, such as might conceivably distinguish it from other 
thinking beings, dictating the form which its thinking ac- 
tivity should take, and in which its particular acts of 



230 A PRIORI AND EMPIRICAL METHODS. [Book IIL 



thought should be combined. It is true there was no occa- 
sion to take this assumption of a possible distinction between 
different beings endowed with a like capacity of thought for 
anything more than a fiction, which served to illustrate the 
truth that no adequate basis of human knowledge is to be 
found in the mere abstract fact of consciousness (cogitatio), 
but only in definite and concrete forms of it which at the 
same time are in fact shared by all minds in common. 
Nevertheless when once the conceivability of such a dis- 
tinction had been admitted, it was no longer possible to 
resist the question what would be the result if it were ac- 
cepted as real ? And then the two sides of the Cartesian 
conception, the a priori character of the Ideas and their 
truth, parted asunder. To each individual that must neces- 
sarily appear to be truth which follows from the laws of its 
own nature ; and so if each is furnished at birth with a 
stock of Ideas in the way supposed, then it is a mere act of 
faith, a faith quite irrational however firmly held, to imagine 
that the Ideas which are allotted to mankind contain a 
higher measure of truth than those which it may be force 
themselves with a no less convincing evidence and with a 
divergent message upon beings of a different constitution. 
It will be seen that such doubts are justifiable not only 
when we contrast the general sum of our knowledge with an 
objective world of existence of which it is supposed to be 
the copy, but even when — a thing which seems still more 
unavoidable — we insist on counting that only as truth which 
appears to all minds equally necessary, as distinguished 
from that which presents itself differently to different 
minds. This is the point from which the modern polemic 
against the Ideas takes its start, insisting that if our Ideas 
are innate they have no claim to truth, and that such a 
claim can only be allowed if they are regarded as inde- 
pendent of the possible differences between one mind and 
another and dependent only on the nature of a world of 
objects common to them all. 



Chap. III.] CONDITIONS OF INTERACTION. 231 

325. Before we enter upon the arguments on either side 
in regard to the questions which are here raised, it is in- 
cumbent on us to realise that we have now arrived at a 
point at which, instead of the unavowed assumptions to 
which we are in the habit of surrendering ourselves, it 
becomes necessary to make one express assumption which 
we admit in plain terms to be such. No enquiry of this 
nature can establish its conclusions, whatever they may be, 
without making some kind of assumption by the way as to 
the mode in which the object of knowledge may be con- 
ceived as operating upon the subject which apprehends it. 
Let us, instead of thus assuming our postulate by the way, 
place it at the head of our enquiry, in the shape in which 
the varied experience of the human mind has taught us to 
formulate it. Wherever between two elements A and B of 
whatever kind any event which we call the influence of A 
upon B occurs, such influence never consists in a con- 
stituent element, or predicate, or state a separating itself 
from A to which it belonged, and just as it is, and without 
undergoing any change, passing over to B, to attach itself 
thenceforth to this new object, or be adopted by it, or 
become one of its states (however we like to phrase it) ; 
what happens is, that a, the property residing, or change 
arising in A, becomes the cause by reason of which, given 
a relation C already established or coming for the first 
time into play between A and B, B also is necessitated in 
its turn to evolve out of its own nature and as a part of 
itself its new state b. 

How this necessary connexion between the states of A 
and B is brought about, how it happens that B is necessi- 
tated to follow the changes of A, what again the relation (7, 
which may be constant or may vary in different cases, but 
which is essential to the production of the effect in question, 
consists in ; — all these questions, as well as the preliminary 
one whether they admit of an answer at all, may be left 
outside our present enquiry; for us the abstract principle 



232 A PRIORI AND EMPIRICAL METHODS. [Book III. 



enunciated is sufficient, no matter what the mode in which 
it is realised in fact. That principle however gives us this 
result, that the form of the effect b can never be independent 
of the nature of the object B which experiences it ; it 
changes with that object ; and the same relation C which 
obtained between A and B, will as between A and B\ pro- 
duce in B l a new effect b 1 quite distinct from b. As little is 
the effect b independent of the nature of the active agency 
A or of the relation C ; it changes with both • if A 1 instead 
of A enters with B into the relation C, it will become ß, and 
ß 1 if B and A enter into the relation C l . But all these 
different results b, b l , ß, ß 1 will make up in themselves a 
complete series of events which are only possible in B, and 
A and C are only to be regarded as exciting causes, deter- 
mining which of the many effects of which the nature of B 
is susceptible are to be realised at a given moment, and in 
what order they are to come about. If we like to apply 
here the favourite designations, receptivity and spontaneity, 
we may say that every object is receptive of various kinds 
of stimuli to its spontaneity, and never operates sponta- 
neously without such stimulus. 

326. The operation of objects of knowledge upon a sub- 
ject apprehending them comes under this general principle. 
Every assumption, to begin with, is wholly inadmissible 
which places the origin of our knowledge exclusively in the 
object : a very little attention will discover to us that even 
in the ' tabula rasa ' to which the receptive soul has been 
compared, or in the wax, which it has been supposed to 
resemble in being a mere recipient of impressions, a spon- 
taneous reaction of the recipient subject is indispensable. 
Only because the tablet by virtue of certain modes of 
operation peculiar to its nature and consistence retains the 
coloured points and prevents them running into each other, 
only because the wax with its cohesive elements presents 
the properties of an unelastic body readily receptive of the 
stamp and capable of retaining it — only by virtue of this 



Chap, in.] MERE RECEPTIVITY IMPOSSIBIE. 



233 



peculiar nature of theirs are the tablet and the wax adapted 
to receive the colours or the stamp impressed upon them ; 
an object which presented no such qualities of its own to 
meet the stimulus from without would not possess so much 
as the character of pure receptivity ascribed to it. 

Further it is necessary clearly to understand, that in an 
act of knowledge the direct contribution from the side of 
the object may be absent, but never that which is furnished 
by the subject's own nature. For it is conceivable that two 
ideas a and ß, having once arisen in the soul through a 
stimulus from without, should then combine in obedience 
to laws having their source in the constitution of the mind 
alone, and without any renewal of the external stimulus, in 
a new result y ; but it is quite inconceivable that we could 
receive an impression from the w r orld outside with the 
shaping of which our own nature had nothing to do. And 
therefore we cannot assent to the distinction between the 
matter and form of knowledge as it is drawn by Kant. The 
idea is indeed perfectly just, but he formulates it inaccurately 
when he ascribes the entire content to experience and the 
form alone to the innate activity of the mind. Kant was 
well aware of the fact which we are here emphasising, that 
even the simplest sensations, which in the strictest sense 
furnish the original content of all our perceptions, do not 
come to us ready made from outside, but on the contrary 
(if we are to hold to the conception of an external world) 
can only be considered as reactions of our own nature of 
combined sense and intellect in response to the stimuli 
coming from that world. They are the a priori capacities 
of experiencing sensation having their seat in ourselves 
which the external forces do indeed summon into actual 
existence in a definite order, but never transmit simply to 
us ready made. And when we pass to the composite result 
of these simple elements, the image of a particular form 
presented in space, the succession in time of the notes in a 
melody, or of a series of events, these too, in every particular 



234 A PRIORI AND EMPIRICAL METHODS. [Book III. 



and detail of the picture, are no whit less the product of 
the thinking subject, no whit less therefore a priori. For 
even if we assumed that things exist in a real extended 
space or occur in a real order of time in the same positions 
or in the same order in which we thereupon apprehend 
them, even then our temporal and spatial idea of them 
would be something quite different from their temporal and 
spatial existence ; we could not manage to bring our ideas a, 
ß, y, into the same order as obtains among their objective 
causes a, b, c, unless our own nature and the laws of our 
mind enabled and obliged us to do so. 

327. Or do we wish to delude ourselves with words and 
to reply that this trifling business of copying may be taken 
as a matter of course and requires no such labour of re- 
creation as we have attributed to the mind ? But what do 
we mean by this word copy, and how is an image or a pic- 
ture produced ? We will say nothing at present of the eye, 
for which alone after all a picture is a picture, and we will 
ask only what are the conditions which make it possible for 
a mirror to present to the eye the image of any object ? It 
can only so present it by reflecting the rays of light which 
it receives from the object in a fresh direction, w r hile main- 
taining their original arrangement relatively to one another, 
and for this office it is absolutely dependent on the smooth- 
ness and the shape of its surface. It depends on these 
qualities of its own v/hether it scatters the rays in such dis- 
order that no eye can combine them into a picture, or 
whether it so reflects them that although they diverge they 
can still be collected by the eye, or so that by converging 
they compose a real image which becomes visible to the 
eye as a new object. 

But even when all this is done the mirror only supplies 
the stimulus which acts upon the organ of sight similarly to 
the object itself, and can be taken therefore to represent it ; 
but if we ask how it is brought to pass as a result of this 
stimulus that the picture reflected can be seen, we are at 



Chap. III.] 



ACTIVITY IN PERCEPTION. 



235 



once sensible how inapt the comparison of knowledge with 
a copy is. The apprehending consciousness is no resisting 
surface, curved or plane, smooth or rough, nor would it 
gain anything by reflecting rays of light no matter in what 
direction ; it is in itself and its own co-ordinating unity, 
which is not a space, and not a surface, but an activity, 
that it has to combine the separate ideas excited in it into 
the perception of a spatial arrangement, which perception 
again is not itself an order in space but only the idea of 
that order. For even if, as some persons may perhaps 
imagine, the idea of a point to the left were actually placed 
to the left in our consciousness side by side with that of 
another point to the right, and the idea of an upper point 
above that of a lower, still this fact would not by itself give 
us the perception of this fact ; all that this by itself would 
do would be to place us this time really in no better con- 
dition than that of a mirror in which some other mind 
might discover the disposition of the points, but again only 
on the supposition that it succeeded in accomplishing that 
which our own mind had not done, that is to say that it not 
merely received and retained the impression of the rays 
with their order of arrangement as reflected from our mind, 
but also turned those impressions to account, by producing, 
on occasion of them, a co-ordinating perception of that 
order. 

Nothing is left therefore of this inexact comparison except 
the conviction that even the mere perception of a given 
state of things as it really is, is only possible on the as- 
sumption that the perceiving subject is at once enabled and 
compelled by its own nature to combine the excitations 
which reach it from objects into those forms which it is to 
perceive in the objects and which it supposes itself simply 
to receive from them. 

That the case is the same with all the ideas which we 
form as to the inward connexion between one perception 
and another, is a fact to which I need only briefly advert, 



236 A PRIORI AND EMPIRICAL METHODS, [Book III. 

for it is here that the criticism has been most generally 
admitted. It is allowed on all hands that we do not see 
the causal connexion between two events, but that on the 
contrary the idea of such a connexion has to be superadded 
by ourselves to that mere succession of events in time which 
is alone directly perceived ; and the admission of the a 
priori origin of the causal nexus has been used by one 
school of philosophy to establish for it the superior dignity 
of a necessary idea of universal validity, and by another to 
deny it all validity whatever in relation to the world of 
things in our perception of which its origin is not to be 
found. Both the one deduction and the other is unsound. 
In regard to the second I recall once more this simple con- 
sideration ; even if a causal connexion does exist between 
the events of the world outside us, it still could not possibly 
be presented to us as the direct object of a purely receptive 
faculty of perception • the mode in which individual im- 
pressions are connected can never do more than afford a 
stimulus to thought to introduce the conception through its 
own activity, nor can such stimulus actually operate unless 
our intellectual nature is itself necessitated, in order to 
complete and account for the observed combination of im- 
pressions, to supplement it by that conception. 

328. The a priori character however, which we thus 
claim in so broad a sense for our knowledge, is only one 
side of the matter. If we regard all forms of sensible per- 
ception, our intuition of space, our conceptions of thing 
and quality, of cause and effect, lastly the ethical ideas of 
good and evil, as modes of manifestation innate in the 
mind, then and for that reason the ground for this and that 
particular application of them, one necessarily excluding 
another, cannot possibly be found in the mind. In our 
perception of space there are innumerable figures possible, 
but at a given moment we only observe certain definite 
ones ; we are capable of seeing many different colours and 
hearing very various successions of sounds, but we cannot 



Chap. III.] KNOWLEDGE AND EXTERNAL WORLD. 237 



alter the red which we have before us here and now, though 
blue or yellow in the same place would be equally percep- 
tible to us, nor can we substitute for the melody to which 
we are now listening any other of the countless melodies 
which we have heard at other moments ; events follow one 
another independently of us, now forcing us to recognise a 
causal connexion between them, now making such an as- 
sumption impossible ; finally this grouping of the incentives 
which are offered us to the exercise of our a priori faculties 
varies as between one individual and another, and cannot 
therefore have its foundation in the common nature of the 
mind. 

To what it is that we are to attribute them is here in- 
different. It may be that the ordinary opinion in which we 
all acquiesce in practical life, and from which the present 
discussion started, is the true one ; that there does exist a 
world of things outside us, in which we have ourselves our 
assigned places, and which affects us in varying ways 
according to the changes which take place in itself and to 
the different and varying positions which we occupy in it. 
In that case the complex web of ideas which forms itself 
within us, cannot indeed claim the name of truth in the 
sense of presenting to us a real likeness of that which exists 
or occurs in the world of things ; still each several con- 
junction or separation or transformation of the phenomena 
which float before our consciousness, will in its character of 
a consequence bear witness to a definite process of change, 
though it may be of a different order, in the relations of that 
world of things which operates upon us. And we should be 
led to the same conclusion by the rival doctrine of Idealism 
which never becomes natural to us in ordinary life, and is 
recommended solely by arguments which lie purely within 
the field of philosophy. It may be, as this belief supposes, 
that there is no world of things or events outside us, but 
only the appearance of such a world brought about within 
individual minds, and nowhere else, by a single unknown 



238 A PRIORI AND EMPIRICAL METHODS. [Book III. 

power which penetrates them all, and that in such a manner 
that the pictures of the world which different minds seem to 
themselves to see around them, fit in one with another, 
and all are presented to themselves as members, each in its 
own place, of one and the same universe. This theory, like 
the other, has necessarily to admit that the stimulus which 
excites any individual mind to create its particular picture 
of the world, is a stimulus foreign to itself, and at the same 
time not explicable from the universal spiritual nature which 
it shares with all other minds. Wherever it may come 
from, it remains an empirical or a posteriori element in our 
knowledge. And again : every conjunction or separation 
or diversification of the phenomena which so arise in us, 
will point to a distinct occurrence elsewhere, to changes 
taking place, not indeed any longer in the relations of 
manifold external objects, but in the action of that one 
power which creates within us this dream of an external 
world. Here finally as on the former hypothesis it would 
be well worth while to establish by observation and com- 
parison of the phenomena those unchanging laws which 
they follow through all the play of change ; and the accom- 
plishment of that task will still give a knowledge of truth, 
even though there were no means of deciding what is the 
nature of that distinct set of laws obtaining in an unknown 
outer world which are the source of the orderly government 
of our own world within. The view I am here representing 
is in essentials that of Kant, and is one which German 
philosophy ought never to have deserted. But in so doing 
I expressly decline to give any answer to the question last 
alluded to. Let a man believe himself ever so much to 
possess an immediate certainty of the existence or the non- 
existence of an external world of things ; the nature and the 
manner of that existence can still only be unriddled by 
conclusions drawn from phenomena. Here therefore our 
footing must be secured to begin with; we must first 
establish those certain principles which are to determine the 



Chap. III.] 



MEANING OF 'EXPERIENCE: 



239 



judgments we form in regard to the system of this inner 
world, before we can talk of applying the conclusions so 
obtained to the further metaphysical question. 

329. But now supposing that we assume certain truths as 
innate, in the previously accepted sense of the word, whence 
do we arrive at the knowledge of them, unless it be by 
discovering them within us, that is to say, by inward ex- 
perience? So that after all experience will be the sole 
source of all our knowledge ? This criticism has been 
made, and it will be felt prima facie to be as barren as it is 
unanswerable. For certainly to know a truth we must be 
conscious of it, and if we were not conscious of it before, 
then the passage to the knowledge of it is an event which 
we must necessarily live through or experience ; in this 
sense of the word our whole existence is a fact which only 
experience discovers to us. This objection therefore to the 
a priori nature of innate ideas cannot hold ; on the contrary, 
supposing there to be innate ideas, supposing them to exist 
even in the sense of being unceasingly present to conscious- 
ness, still the mind reflecting on them could, to begin with, 
only be aware of their presence as a fact given in its ex- 
perience or its conscious life. Taken then in this broad 
acceptation the conception of experience no longer offers 
occasion for a difference of opinion ; the only point of im- 
portance is as what do we experience the thoughts in 
question ? Do we experience them as innate truths, or as 
matter of experience in that narrower sense, in which they 
indicate in contradistinction to such truths that their origin 
is foreign to the mind itself? With this distinction the 
question about experience seems at first sight to take a more 
urgent form ; if, that is, we go on to ask for marks which 
may distinguish the one of these cases from the other. We 
then find that the impressions which come to us from 
outside are forced upon us and we cannot alter them ; but 
the a priori truths also present themselves as unavoidable 
and unalterable \ that the compulsion in the first case 



240 



A PRIORI AND EMPIRICAL METHODS. [Book III. 



comes from without, and in the second is that of our own 
nature, we may indeed conjecture, but how are we to prove 
it ? The truth however is that if we take the unsophisticated 
intelligence we find that this which to us in the course of 
our methodological investigation was the most important 
fact, is not to it the primary one at all ; the truths in 
question are not matter of experience in respect of their 
alleged quality of being innate in us ; what first strikes us is 
that as a matter of fact what they assert is self-evident, so 
that when once we have had occasion to think of them in 
any particular instance, we see them to be independent 
of any further confirmation through fresh instances, and 
thus independent of experience which might supply such 
instances. And hence universality and necessity have always 
been the two characteristics which have been ascribed to 
a priori knowledge. We understand by the term universality 
that invariably as soon as the subject is thought of the pre- 
dicate which belongs to it appears in self-evident conjunction 
with it ; and again it is in this self-evidence and in nothing 
else that necessity or necessary validity in this sense consists, 
for clearly necessity attaches to universal truths in quite a 
different sense from that in which it belongs to those con- 
junctions of various objects which our changing experience 
brings before us. These objects, it is true, are also pre- 
sented to us in such a way that at the moment in which 
they occur we cannot dissolve the conjunction at our 
pleasure ; but though the content of experience possesses 
necessity in the sense in which every fact which cannot be 
denied does so, still it lacks that perfect self-evidence which 
consists in an inherent connexion of elements which are 
unthinkable apart from each other. 

But after all, what gives us the right to affirm that that 
which may appear to us self-evident at this particular 
moment will appear so equally at every other, that is to 
say, to ascribe to it a universality which can make it a fixed 
principle of judgment in face of a perpetually changing 



Chap. III.] EXPERIENCE INVOLVES PRINCIPLES. 241 



experience ? This question was raised by the early Sceptics 
and led them to declare all general propositions inadmissible. 
And in point of fact, whatever principle we may choose to 
devise to justify us in concluding from the certainty of a 
proposition at the present moment to its certainty for all 
future time, must itself be subject, as a universal principle, 
to the precise suspicion which it was intended to remove. 
Thus we should have no means of assuring ourselves of the 
universal validity of any proposition if we cannot be satisfied 
with the self-evidence with which its content, once thought, 
claims for itself eternal validity in anticipation of experience. 
And it would have to be a matter of consideration that this 
incapacity for attaining to universal truth could not be 
deplored as an infirmity peculiar to the human intelligence ; 
it would be shared by all minds whose experience as being 
developed in time at all resembles ours ; the very truest 
truth which might be innate in such a mind could only 
come into its consciousness at a definite moment, and all 
the self-evidence it might possess for it at that moment 
would not remove the uncertainty whether it would remain 
a necessity of thought in the next. 

330. This result will perhaps be eagerly admitted, and it 
will be urged that it proves the futility of our defence of 
a priori truths \ even when the mind has got them it has no 
means of distinguishing them from the results of experience. 
Or in other words it is only experience which teaches us 
that they have universal validity ; that is to say, when we 
find their self-evidence confirmed by each successive 
attempt to think them, we have not indeed strict proof 
but we have the strongest probability that they are valid 
without exception, and it is to this gradually increasing 
empirical probability that the whole of our knowledge is in 
fact restricted. 

In this there is an element of truth which I shall con- 
sider presently ; but taken as a whole it is a false position. 
If we assume, as this view admits, that the certainty of a 

Logic, Vol. II. R 



242 A PRIORI AND EMPIRICAL METHODS. [Book III. 



given proposition as experienced at one moment does not 
guarantee the experience of its certainty in the next, then 
just because this is so a thousand repetitions will not make 
it a whit more probable in the thousand and first case than 
it was in the second or third. If after a series of cases of 
the connexion of two events a and b unbroken by any 
instance to the contrary, we look for fresh instances with 
constantly increasing confidence, we do so on the strength 
of very definite assumptions. If the connexion of a and b 
is not of such a kind as to make it self-evident the moment 
it is presented to the mind, if its eternal validity is not at 
once apparent, then we explain its constant occurrence by 
the fact that the conditions which might have produced a 
different result have not so far come into operation; that 
they are not likely to do so at any future time we conclude 
after numerous instances of similar experiences on the 
strength of one special assumption and not otherwise, the 
assumption that the course of the universe in general and 
of this part of it to which the events in question belong in 
particular, proceeds in a fixed order, which by examination 
of a sufficient number of instances, becomes discoverable. 
Then, starting from this assumption that a particular set of 
conditions whenever they recur in the future will be equiva- 
lent to what they were when observed in the past, we draw 
our conclusion : given like conditions a like result must 
present itself. If we are wrong in that assumption this will 
mean that we have set up as universal a false generalisation 
concerning a matter of fact, which will be refuted by future 
experience. On the other hand, if our universal principle, 
that under like conditions like consequences follow, is no 
longer to be regarded as really universal, then the entire 
method of logical procedure by which we expect to pass 
from particular experiences to propositions of even probable 
universality, is absolutely baseless and vain. For every time 
we argue from m to m + 1, whether we are undertaking to 
establish a strictly universal or a merely probable con- 



Chap. III.] REAL AND SPURIOUS SELF-EVIDENCE. 243 



elusion, in either case we assume the strict universality of 
that logical principle. 

It is clear therefore that the attempt to derive the entire 
body of general knowledge from experience, that is to say 
from a mere summing up of particular perceptions, breaks 
down. We have invariably to help ourselves out by 
assuming at one point or another some one of those self- 
evident principles, some principle to which when once its 
content has been thought we at once concede with intuitive 
confidence that universal validity to which it makes claim. 

331. Now in practice as a matter of fact there has never 
been any dispute on this point. Mathematical demonstra- 
tions have often been subjected to fresh examination, but 
never with any other object than to establish whether each 
one of the several propositions which made up the chain of 
reasoning was either itself self-evident or was logically 
derived from others which were so. We never set to work 
merely to prove over again the self-evident propositions 
themselves, to see whether some moment may not arrive in 
which their direct contraries, the equality of unequals for 
instance, or that the part is greater than the whole, would 
be equally self-evident ; and even supposing so unexpected 
an event had on some occasion occurred, no one would 
have doubted that there was an error somewhere, which 
could only be attributed to an oversight in the calculation. 
On the other hand much difference of opinion does exist as 
to the extent of these universal and self-evident truths, and 
here we are brought in view of that element of truth which 
I could not help allowing above, in the theory just combated. 
I by no means intend however to imply that experience as 
such could help us to establish what holds universally not 
merely as a universal fact, but as a self-evident and neces- 
sary truth ; on the contrary it is precisely experience with 
its repeatedly recurring uniformities which at last deludes 
us into taking for necessary and self-evident truth, that 
which is merely matter of fact, or not even that. 

r 2 



244 A PRIORI AND EMPIRICAL METHODS. [Book III 

I have spoken before of the delusive certainty which 
many principles assume, merely because our limited experi- 
ence has constantly presented them to us without any 
instance to the contrary. The psychological association 
which establishes itself under such conditions between the 
ideas a and representing two events which have constantly 
followed each other, very soon assumes the appearance of 
a self-evident connexion in fact between the contents of the 
ideas so presented. I observed then that the attempt to 
think the direct contradictory of a proposition which has 
come to be thus self-evident may serve sometimes to dispel 
the illusion, and we then find to our astonishment that a 
hypothesis which contradicts our apparently self-evident 
proposition presents no difficulty to thought, that it is just 
as much thinkable as the other, and that accordingly the 
certainty which we ascribed to our belief cannot depend 
upon any universal self-evident connexion in its content. 
I was obliged however even then to add that this aitempt 
to think the contradictory will not always be a decisive test ; 
the influences of previous experience which nullify its value 
are in fact very various. If we could be certain, in apply- 
ing it to any proposition, that we have not only determined 
with perfect exactness, with nothing lacking and nothing 
over, the meaning of the subject a, and the predicate and 
also of the copula c or whatever the connexion may be 
which we wish to establish between them, but also that in 
the final decision as to whether that relation c which we 
have established is self-evident or not, we have been guided 
by no sort of consideration save the fixed meaning of the 
three conceptions ; then undoubtedly we should all agree 
in our conclusions, positive and negative alike. And wher- 
ever these conditions are susceptible of fulfilment, as is 
the case in mathematics, such agreement is in fact found. 
The complex notions on the contrary of real objects are 
very far from admitting the same exactness of analysis, 
and every reasonable man looks for results in this sphere 



Chap. III.] 



CRITICISM OF PREJUDICES. 



245 



only from experience or rather from the accurate mani- 
pulation of our experiences. Finally those simplest and 
most universal conceptions and principles to which we 
should desire to subordinate that manipulation, would 
unquestionably admit of the highest degree of such ac- 
curacy, did not the influence of past experiences come 
in the way. We certainly intend something very simple 
and definite when we use the words, being, thing, cause, 
force, effect, matter; but in our use of any one of them 
we are commonly determined by our limited circle of experi- 
ence or our favourite study or pursuit. Thus we are led 
on the one hand to apply them only to a fraction of the 
subject-matter which we in fact hold that they ought entirely 
to dominate, and yet on the other hand to bring them into 
a variety of connexions which are not indeed impossible to 
them but still do not essentially belong to them. Thus we 
might perhaps if we were required to define one of these 
conceptions agree in our definitions, yet the ways in which 
we actually look at its meaning might be different enough, 
as different at all events as in the case of the same objects 
seen in different lights. Now all these unanalysed side- 
thoughts, the emotional suggestions and the wishes which 
thus attach themselves unawares to the object of thought, 
and give it its characteristic colouring, dispose us to find 
the certainty of self-evidence in predicates which we should 
not be warranted from the nature of the object alone in 
applying to it at all. This is at once the value and the 
danger of experience ; except as suggested by experience 
the universal principles of our judgment cannot be pre- 
sented to consciousness at all ; but as thus occasioned they 
are at the same time subject to one-sidedness, deficiencies 
in one direction, superfluities in another, from which later 
reflexion has much ado to purify them. Here begins a 
work of criticism which has to be unremittingly pursued ; 
the useful labour of investigating the psychological origin of 
the particular form which these conceptions have come to 



246 A PRIORI AND EMPIRICAL METHODS. [Book III. 

assume in our consciousness ; the object being not so much 
to show how all certainty and truth arises little by little out 
of the deliverances of experience, as, on the contrary, to 
make it clear how much foreign matter due merely to the 
peculiarities of the instances observed, has incrusted itself 
upon the substance of those original truths, truths which, if 
once they were seen in their simplicity and purity, would be 
not only recognised as necessary and self-evident, but would 
prove so in all their applications. 

332. Such a criticism of prejudices, as I may shortly call 
it, cannot I conceive be conducted otherwise than piece by 
piece in connexion with definite problems which offer them- 
selves for solution ; for it is only difficulties which rise upon 
us in working out individual problems, which lead us to 
suspect the soundness of our principles and to cast about 
for the sources of the errors we have fallen into. I refrain 
therefore from entering into the subject here in detail ; on 
the other hand it is necessary that I should vindicate the 
method I have thus far pursued as against the opposite 
theory, which not content with freeing the primary truths 
by this process of psychological analysis from the erroneous 
side-thoughts which have grown up about them, aims further 
at giving a systematic explanation of the nature of thought 
and demonstrating the validity of its first principles. I 
have maintained the opinion throughout my work that 
Logic cannot derive any serious advantage from a discus- 
sion of the conditions under which thought as a psychical 
process comes about. The significance of logical forms is 
to be found in the meaning and purport of the connexions 
into which the content of our world of ideas ought to be 
brought ; that is to say in the utterances of thought or the 
laws which it imposes, after or during the act of thinking, 
not in those productive conditions of thought itself which 
lie behind. Conditions of this kind there must certainly be, 
not only those conditions of a psychical mechanism which 
determine at every single moment every single one of its 



Chap. III.] LIMITS OF PSYCHOLOGICAL ANALYSIS. 247 

motions, just as every feature in an event of external nature 
is determined by the physical conditions which are given at 
the moment of its occurrence, — but more than this, the 
necessity with which, speaking generally, thought follows 
unawares those logical rules of its procedure which later 
reflexion formulates into consciously apprehended prin- 
ciples, must be an unavoidable consequence of the nature 
of the mind itself, which it belongs to Psychology to inves- 
tigate. But if we knew all that we could desire to know on 
the subject, it would still be a delusion to suppose that we 
should be thereby any the better able to judge of the truth 
of our logical principles ; on the contrary the validity of 
those principles themselves would still be the necessary 
postulate without which the successful enquiry into their 
psychological history could not have been undertaken at 
all. 

To touch here for the last time upon this logical circle 
which has wearied us so often already; it must be clear 
enough that no sensational or empirical theory of the origin 
of thought and knowledge can possibly either prove or dis- 
prove the principle of identity or excluded middle ; in every 
step of the argument it needs them both. As little can it 
be left to such a theory either to establish or to destroy the 
validity of the law of causation. For every attempt to re- 
duce our application of it in the field of experience to the 
association and reproduction of ideas presupposes its 
validity in another form in relation to the interaction of 
psychical states; so that it can neither be accepted nor 
rejected unless its validity be established to begin with — a 
premiss from which certainly the rejection of it could only 
be arrived at by a very curious sort of logical suicide. No- 
thing then remains but to restrict this psychological analysis 
to the task of showing how truths which have their own 
validity in themselves find realisation in thought and for 
thought, regarded as a psychical process, as rules of its pro- 
cedure which it follows unawares. 



248 A PRIORI AND EMPIRICAL METHODS. [Book III. 

333. And now I should like to make clear that of all 
that we might wish to know in this direction we in fact 
know nothing at all, and that Logic would have to renounce 
for a long time yet any profounder understanding of the 
operations of thought if she had to look for it in the psycho- 
logical analysis of their origin. In the works of the sensa- 
tional school, which have been produced in such numbers 
and such variety on the model of Locke's Essay — which is 
here unrivalled — and of Condillac's bold venture, I can find 
nothing that answers in a general sense to this requirement. 
Regarded as a criticism of the prejudices of human thought, 
Locke's work has enjoyed the full measure of influence in 
the development of modern philosophy to which the wide 
horizon which it opened and the keenness of its analysis 
entitled it. But in dealing with all the variety of those 
inner processes of the mind, which he undertakes to criti- 
cise, Locke has no other instrument to apply but ' common 
sense,' a faculty which, versed in the criticism of the course 
of events in the outward world, imagines that the very re- 
spectable and probable but quite unsystematic maxims 
there acquired are sufficient to meet all emergencies. It is 
more to my purpose at present to consider the attempts 
which have been made in this direction in German philo- 
sophy. When we speak of explaining any set of processes, 
and regret its non-accomplishment, we think, as the type of 
the wished-for ideal, of the body of the natural sciences. 
By the strict observance of the laws of thought and the 
careful application of them to the results of exact observa- 
tion, natural science has succeeded in arriving at a small 
number of original facts from the interaction of which ex- 
ceedingly various phenomena can be shown to follow with 
logical necessity. A series of happy inspirations 1 have 
within quite recent times added to this domain a portion of 
the inner life of the soul, at least in regard to the depend- 
ence of sensations upon external stimuli. And this result 

3 [For an account of these investigations, see ' Metaphysic/ § 258.] 



Chap. III.] 



MODERN PS YCHOLOG Y. 



249 



was due not to attempts to construct the entirely peculiar 
set of events which we call psychical out of physical pro- 
cesses, which can never be brought into any comparison 
with them ; but to investigations of which the aim has been 
simply to apply exact quantitative determinations to the 
members of the two series which the order of nature does 
actually unite together, though in a manner unknown to us, 
and from the pairs of correlated values thus ascertained to 
develop the laws of their correspondence. And previously 
to these enquiries a valuable attempt had been already 
made 1 , not indeed resting on the exact observation of 
special facts, but upon hypotheses suggested by experience 
generally, to bring the purely inward phenomena of mental 
life under a mechanical theory of their origin. At the same 
time all these achievements which have given the psychology 
of the present day a very great superiority over the views of 
earlier times, do not reach those obscure regions of enquiry, 
the illumination of which might open new paths to Logic. 
They merely instruct us concerning the interaction of dif- 
ferent psychical states to which measurement has been 
applied, in regard to the changes they severally undergo 
when brought into connexion with each other, and thus in 
regard also to the total state of the soul at any moment, 
considered simply as the mechanical result of all these re- 
ciprocal influences. But they do not equally explain the 
fresh reactions to which the soul is stimulated by each one 
of these states of itself as they thus arise, and which are not 
calculable consequences of certain quantitative relations in 
the co-operating conditions, but depend, in obedience to a 
necessity of a wholly different order, shall we say a dialecti- 
cal or teleological necessity, upon the meaning or the idea 
which the soul is destined to realise. 

The investigation of external nature leaves questions of 
this sort behind, but for its purposes it does not need to 
answer them. In what way it happens, by what means it 
1 [An allusion to Herbart, see * Metaphysic,' §§ 269, 270.] 



250 A PRIORI AND EMPIRICAL METHODS. [Book III. 



is brought about, or to what purpose it tends, that particles 
of matter attract each other with a force determined by 
their distance, are questions which may be left undecided. 
When once the law of this reciprocal influence is ascer- 
tained, it can be reckoned as a constant element in the 
course of nature, that is to say in the present case as an 
element into the determination of whose variations in each 
several instance the given circumstances enter. The more 
we succeed in reducing all natural processes to homogeneous 
motive forces of this kind, the more possible will it become 
to construct even the form of every single natural event out 
of the conditions which occasion it. This would all be 
altered if the natural sciences had cause to suppose that 
the material elements which had hitherto been regarded as 
unchangeable, experienced under the operation of forces of 
this kind certain inner changes which had the effect of 
stimulating them to wholly new modes of reaction, giving 
them a new influence in the play of events. No doubt 
those new influences so far as they operated to bring about 
changes in the physical surroundings could still be directly 
connected with the ascertainable outward conditions under 
which they arise, or, to express it in general terms, they 
could be regarded as functions of the conditions ; and thus 
there would be apparently no interruption in the continuity 
of the scientific construction, only an increased difficulty in 
carrying it out. But in point of fact a breach of continuity 
would certainly have taken place. For the simple fact that 
given a certain set of physical conditions m a new mode of 
operation \x will make its appearance and given another set 
n a second new result v, would remain after all a new datum, 
a fact known indeed from experience, but not to be derived 
analytically as a necessary and self-evident consequence 
from the physical conditions given. 

Now the case in which we find ourselves in regard to the 
present question is analogous to this. All the mental pro- 
cesses which psychology teaches us are necessary presuppo- 



Chap, in.] THOUGHT AND PS YCHICAL MECHANISM. 2 5 1 



sitions for the realisation of any act of thought, are merely 
the conditions m or n which give occasion to the logical 
reactions jjl and v to present themselves. They cannot ex- 
plain the fact that fx and v do thus appear upon the scene, 
nor again do we find in this fact in itself the least explana- 
tion of the further relations of constantly increasing com- 
plexity which thought establishes between its /x and v or 
other of the elementary products of its activity. 

I should dwell upon this point further were it not that 
the subject of the following section will oblige me in any 
case to call attention later on in detail to the deep gulf 
which remains unfilled between the psychical mechanism 
and thought ; I content myself here with the expression of 
my conviction that all logical reactions of the mind have to 
be conceived as a connected whole, as expressions of a 
single tendency whose separate utterances can in so far as 
their meaning is concerned be apprehended and arranged 
in an intelligible series, but in their origin as psychical pro- 
cesses remain wholly incomprehensible. It is an illusion \ 
in psychology and a corruption of logic to take the con- 
ditions which occasion the logical operations of thought for 
the operations themselves. There is only one delusion 
more desperate still, — to imagine that a complete physical 
theory of the nervous system will explain that which is 
itself the condition of any theory being possible at all 



CHAPTER IV. 



Real and Formal Significance of Logical Acts, 



334. Facts of perception we acknowledge without ques- 
tion ; our misgivings begin with the interpretations of those 
facts by discursive thought, more especially when we con- 
sider the protracted and intricate web of ideas which thought 
spins in abstraction from the facts of sense, yet always with 
the expectation of reaching a final result which perception 
will confirm. Thought as an activity or movement of the 
soul follows laws of the soul's own nature ; will these laws 
which it necessarily follows in the connexion of its ideas, 
lead to the same result as that which the real chain of 
events brings round ? Will the outcome of the process of 
thought, when at the close of it we turn once more to the 
facts, be found in agreement with the actual results which 
the course of nature has produced ? And if on the whole 
we consider it improbable that thought and being, which it 
is natural for us to regard as made for one another, should 
be entirely divorced, are we also to suppose that every 
single step taken by thought answers to some aspect of that 
which actually takes place in the development of the things 
thought about ? Such are the doubts which give rise to the 
theory of the purely formal or subjective validity of thought. 
That theory is perfectly clear in what it affirms ; the logical 
forms and the laws of their application are the conditions 
through the fulfilment of which thought satisfies its own 



FORM DEPENDENT ON MATTER. 



253 



requirements, and brings the connexion of its ideas with one 
another into that form, which for it, for thought itself, is 
truth ; but it is not at all clear what is the relation — though 
some such relation cannot be dispensed with — in which 
these forms and laws stand to the content which they do 
not create but find, and from the manipulation of which 
alone after all that which is truth for thought draws its 
material. 

Can an object, we ask, be brought into forms to which it 
is not adapted? Or even supposing that we are able to 
force our material into a form which it does not naturally 
assume, still must there not be some quality in the material 
which at all events makes such an operation possible? 
Must not every given subject-matter therefore, which 
thought casts into its own forms, possess some relation 
and affinity to those forms, of which the most we can say 
is that it may be misused ? Finally, must not this assump- 
tion hold as regards every single logical operation ? Not 
one of these could be carried out even as a mere subjective 
process of thought, unless the object upon which it is 
exercised contained in itself some characteristic which 
invited or at least allowed it. Now we know that the 
distrust of thought spoken of above, does not find con- 
firmation in experience in the universal sense we dreaded. 
However wrong we may go in protracted chains of reason- 
ing, daily life shows how well our conclusions taken in the 
average agree with the actual course of events. Why should 
we not hold fast to that confidence in the veracity of 
thought which is the natural attitude of our minds before 
scepticism disturbs them ? Why not mount a step higher 
still, and regard the objective 1 content of our world of ideas 
as bound by no other laws than those which thought 
imposes on it ? Then we should need nothing more than 
careful attention to the subtle and intricate logical processes 
of the mind, to find reflected there as in a mirror the real 
1 [' Der sachliche Inhalt des Vorstellens.*] 



254 REAL AND FORMAL VALUE OF THOUGHT. [Book III. 



or objective forms in which all existence appropriately 
developes. 

In this way the belief grows up in a Real significance 
of thought, a belief which in its more general features 
appears in the history of the human mind earlier than its 
opponent, but which stated in these explicit terms and in 
this thorough-going form, is a product of recent times. 
Between this and the opposite theory the history of philo- 
sophy has a long controversy to recount. We cannot 
decide it by placing the logical forms and laws side by side 
with those of real existences and events and comparing the 
one with the other, for we have no knowledge of the latter 
in which thought is not already present and operative. 
But we can ask what is the judgment of thought itself on 
its own operations, and how far it pronounces the forms 
which as a psychical movement of the thinking subject it is 
constrained to assume, to be a determination belonging to 
the object-matter upon which it operates. 

335. To whatever act of thought we direct our attention 
we never find that it consists in the mere presence of two 
ideas a and b in the same consciousness, but always in what 
we call a Relation of one idea to the other. After this 
relation has been established, it can in its turn be conceived 
as a third idea C, but in such case C is neither on the one 
hand homogeneous with a and fr 9 nor is it a mere mechani- 
cal effect of interactions which in accordance with some 
definite law have taken place between the two as psychical 
processes with definite magnitudes and definitely various 
natures. We may take as the simplest examples of what I 
mean the identification and the distinction of two ideal 
contents. If we assume a and a identical 1 with each other, 
then unquestionably the idea a is present twice over in our 
mind, but the only result to which this circumstance can 

1 [' Gleich,' i.e. the same both in quantity and quality. Neither 
' equal ' nor * like' fully render this meaning. Cp. ' Metaphysic/ § 19, 
note.] 



Cbap. IV.] 



COMPARISON. 



255 



lead us on mechanical analogies will be either that the two 
ideas must count as one because they exactly cover each 
other, or that as similar affections of the soul they will 
become fused into a third idea of greater strength, or that 
they simply remain apart without any result at all. But 
that which we call the comparison 1 of them, which leads to 
the idea of their identity C, consists neither in the mere fact 
of their co-existence, nor in their fusion ; it is a new and 
essentially single act of the soul, in which the soul holds the 
two ideas side by side, passes from one to the other, and is 
conscious of experiencing no change in its condition or in 
the mode of its action during or by reason of that passage 
from the one idea to the other. 

Again : let us compare two different ideas a and red 
and yellow. Two external stimuli, which acting by them- 
selves would have awakened severally one of the two 
sensations, might acting simultaneously coalesce in the 
nerve, through which they propagate themselves still as 
physical states, into a third excitation intermediate between 
the two so as to occasion in the soul only a third simple 
sensation. But two ideas which have once arisen as ideas 
in the soul, never experience this sort of fusion. If it were 
to occur, if the distinctive existence of the two ideas were to 
vanish, all opportunity and possibility of comparison, and 
therewith as a remoter consequence, all possibility of thought 
and knowledge, would vanish also. For clearly all relation 
depends upon the preservation in consciousness of the 
different contents unfalsified by any interactions of one 
upon the other; the single undivided energy of thought 
which is to comprehend them must find them as they are in 
themselves, so that passing to and fro between them it may 
be conscious of the change which arises in its own condition 
in the transition. 

In using this language I am fully aware that it may be 

1 [' Vergleichung.' The emphasis on the connexion of 1 Ver- 
gleichung s with ' gleich ' cannot be rendered. J 



256 REAL AND FORMAL VALUE OF THOUGHT. [Book III. 

fairly objected that my designation of the energy in question 
contains mere descriptions which cannot be embodied in a 
construction. But this is exactly the point upon which a 
clear understanding is essential, — that the intellectual pro- 
cesses upon which all thought depends do bear no sort of 
resemblance to those physical events on the analogy of 
which such an objection would like to see them modelled. 
An activity which cannot be said simply to be a movement 
but which executes a movement, which relates itself to two 
objects without introducing any change into them, which 
finally becomes conscious of the direction and the length 
of the path it has travelled by the differences which it 
experiences in its own states, — such an activity cannot be 
brought under the ordinary category of unchanging elements 
with changing relations, or of the equality of action and 
reaction ; and yet at the same time it is something whose 
reality we all feel ; it in fact and nothing else is the instru- 
ment by means of which we accomplish those much admired 
constructions which we would fain apply to it. These 
characteristic peculiarities we have simply to acknowledge, 
and to look for a new set of conceptions which may enable 
us to formulate them without falsifying their nature, an 
order of conceptions which are still a desideratum in philo- 
sophy, and which I by no means consider my own very- 
incomplete formulae to have supplied. 

336. In the instances taken above, a and b, red and red, 
or red and yellow, were objects directly given in perception. 
The ideas of identity or difference C which we obtained as 
the result of the act of relation introduced by the mind, are 
no longer of this character. As a relation of one to the 
other, the identity of a with a, or the difference betivee?i a 
and b, they cannot be really thought without at the same 
time recalling on the one hand the ideas of a and b, which 
form the terms in the relation, and on the other that move- 
ment of thought which carried us over from the one to the 
other. Thus every time we use the term identity or 



Chap. IV.] PROCESS AND PRODUCT OF THOUGHT. 257 



difference we are called on to renew once more all those 
operations of thought through which alone it is possible to 
use them with a meaning ; but when we express the final 
result which we wish to produce by the process of thought, 
by saying that a is the same as <z, or a is different from fr, we 
are implying that the objective knowledge which it was our 
object to arrive at lies entirely and exclusively in this final 
step of the completed comparison. It is not to a and b that 
we ascribe the movement backwards and forwards between 
them through which we discovered their relation to each 
other ) this movement is merely a psychical process, without 
which indeed our result could neither be obtained in the 
first instance nor repeated afterwards in memory, but which 
has nevertheless to be abstracted from the real significance 
of the act of thought to which it ministered, as a scaffolding 
is withdrawn when the building is completed. Thus we see 1 
at once in an example of the simplest possible kind the 
antithesis between the merely formal significance of an act 
of thought and the real significance of its product. Before 
I follow up this line of thought further I wish to advert to 
two sets of processes which add a confirmation on a large 
scale to the conclusions which we have seen suggested by a 
particular instance. 

In the first place we receive the sensible perceptions from 
which thought starts almost without exception under the 
form of space, — in spatial shape, arrangement or relations ; 
hence we come to apply terms of space symbolically to 
every sort of complex relation in order to give it that 
vividness to the imagination in which it would otherwise be 
deficient. We represent ideas of difference by terms of 
distance, distance long and short, in this direction and that ; 
the multiplicity of what is the same by distribution at 
different points of space ; the self-identity of unity 1 by the 
notion of an unchanging place which we assign to the idea in 
question whenever we think it ; lastly we find it difficult to 
1 [' Identität des Einen mit sich selbst.' 

Logic, Vol. II. S 



258 REAL AND FORMAL VALUE OF THOUGHT. [Book III. 

make our conceptions clear, wherever the manifold orders 
of relation which present themselves to thought are such as 
the formulae derived from space are inadequate to express. 
And yet for all this we are conscious that these formulae do 
not reach the heart of the matter ; all these symbols are, we 
are aware, mere subjective aids to the understanding, con- 
venient paths for thought which has to travel up and down 
to reach its goal C, which is in itself wholly distinct from 
them ; what we mean is independent of the mode in which 
we figure it. 

Secondly we are accustomed to clothe our thoughts in 
speech, and even in the silent processes of thought it has 
long become habitual to us to call up the appropriate words 
before the mind ; perception, recollection, expectation, hardly 
reach perfect clearness until we have found adequate ex- 
pressions for them in spoken propositions. The advantage 
thus gained is not in its own nature dependent on speech 
and its sounds, but rather on an inward act of analysis and 
combination which would remain the same if it employed 
other forms of communication ; still in point of fact, now 
that speech is there for the purpose, it is undoubtedly the 
case that the forms which the processes of thought assume 
and the facility with which they are conducted are dependent 
upon the means which speech provides, and thus present 
even national differences, when many and various causes 
have combined to render the formation and syntax of 
different languages dissimilar. Thus the logical meaning 
of a given proposition is indeed in itself independent of the 
form in which language expresses it ; but in practice all 
human thought is compelled to represent its meaning by 
separations, combinations, and readjustments of those ideas 
which the growth of language has attached to single words. 
It is only in this its discursive character, in contradistinction 
to Perception, that thought is a psychical fact. It is in this 
character also that it has been the subject of our logical 
treatise. Logic has never concerned itself with a thought 



Chap. IV.] THE PRESENTATION OF IDEAS. 259 

which did not make its various ideas, one after another, the 
object of its attention, which did not move amongst them 
comparing and relating them to each other, which did not 
symbolise abstract ideas by spatial images, which finally did 
not express its thoughts in the forms and constructions of a 
language. We must expect therefore to find in what we call 
logical operations, logical forms and laws, a considerable 
amount of purely formal apparatus which although indis- 
pensable to the exercise of thought, yet lacks that Real 
significance which for the ultimate results of its activity 
thought does undoubtedly claim. 

337. Let us now return to consider this result. When in 
comparing a and b we are conscious of a change C which 
we experience in passing from one to the other, there is no 
doubt that C must depend upon the nature of the two 
terms of the relation, for it would alter and become C 1 if 
they were replaced by c and d. At the same time the con- 
nexion of C with that objective relation seems to be one of 
dependence merely, and not to consist in being an identical 
copy of it ; as a subjective excitation in us it fails short of 
the objective reality towards which knowledge is directed. 
I should not advert to so subtle a refinement of criticism 
were it not that it gives me an opportunity to return once 
more to the difficult subject of the nature of the act which 
presents ideas. The act of presentation is not that which 
it presents, the idea is not that which it means. And this 
not merely in the obvious sense that neither the one nor 
the other is the fact presented : but I mean that even the 
very simplest ideas, the content of which can only exist in 
thought and is not a thing, have not their content as their 
own predicate ; the idea of yellow is not yellow, the idea of 
triangularity is not itself triangular, or the idea of timidity 
timid, or the idea of a half half as large as that of the whole. 
At the same time the act of presentation is not so com- 
pletely separable from its content, that it could be, or occur, 
or experience change by itself; it is only in as far as it pre- 

s 2 



26o REAL AND FORMAL VALUE OF THOUGHT, [Book III. 

sents that which, itself, it is not; it changes only in ex- 
changing one of these contents for another. Thus even the 
change of which it becomes conscious in its own condition 
can only consist* in a change in the contents presented, 
which with its single activity it comprehends and compares ; 
it cannot be sought in an affection of a wholly different 
character which the mind experiences merely as an after 
result of the stimulus given it by those contents, and which 
becomes observable to consciousness apart from those con- 
tents as an idea C having no resemblance to their own 
relation. He who finds red and yellow to a certain extent 
different yet akin, becomes conscious no doubt of those two 
relations only by help of the changes which he himself as a 
subject of ideas experiences in the transition from the idea 
of the one to that of the other, but at the same time he 
never entertains the apprehension that the relation of red 
and yellow may be something quite different in itself from 
that of the affections which they occasion in him, that red 
for instance may be in itself exactly like yellow and only 
appear to us different from it, or again that in reality there 
is a greater difference between them than we know and that 
their apparent affinity is an appearance only. Such scepti- 
cism might not be groundless if the question was one con- 
cerning the relation of our world of thought to a world of 
things assumed to be external to it, but so long as we are 
considering not this external world, but our own ideas, we 
never doubt that the relations of likeness and difference 
which we experience in the comparison of them, on the 
part of our presentative susceptibility, signify at the same 
time an objective relation on the part of those contents 
which our ideas present to us. 

338. But now after all how is this in strictness possible ? 
^ How can the propositions £ a is the same as a] and 1 a is 
different from express an objective relation, which, as 
objective, would subsist independently of our thought, and 
which thought could only discover or recognise ? We may 



Chap. IV.] c EXISTENCE ' OF RELA TIONS. 2 6 1 

suppose ourselves to know what we mean by a self-existent 
identity of a with <z, but what are we to make of a self- 
existent distinction between a and b ? And what objective 
relation can correspond to this ' between,' to which we only 
attach a meaning so long as it suggests to us the distance in 
space which we^ in comparing a with b, interpolated by way 
of metaphor for the purpose of holding the two apart, and 
at the same time as a connecting path on which our mind 
might be able to travel from one to the other ? Or other- 
wise expressed : difference being neither the predicate of a 
taken by itself nor of b taken by itself, of what is it the pre- 
dicate ? And if it has a meaning only so far as a and b 
have been brought into relation to each other, what objec- 
tive connexion, we must then ask, obtains between them, if 
we consider the relating activity through which we have 
conjoined them in our consciousness as not being exercised? 
Many errors in ancient Dialectic were occasioned by the 
fact that these questions were ignored. Attributes which 
can only belong to things in the reciprocal relation which 
our combining thought establishes between them, were pre- 
dicated of them, not without violence to the logical imagi- 
nation, singly and by themselves. In order that a and b 
might be represented as different, without thought being 
required to establish the difference, the attribute was as- 
cribed to each separately of being in itself a erepov, and the 
act of comparison with a second thing, which alone gives 
any meaning to the term, was to be left wholly out of 
account. The negation which thought, comparing and 
distinguishing, expresses in the proposition * a is not b ' was 
then treated as a positive predicate of a as such, the nega- 
tived term b being dropped out. That is to say it was 
treated as a not-being which yet is, and became thus cre- 
dited with a reality of its own; and this confusion was 
reckoned an important and profound discovery. If b is 
less than a and greater than c, it was a riddle which much 
vexed philosophers, how the two predicates, less and greater, 



2Ö2 REAL AND FORMAL VALUE OF THOUGHT. [Book III. 

which, once separated from the terms of the relation to 
which they belonged, stood in direct opposition to each 
other, could be associated in the same b. 

It would be a task not without interest from many points 
of view to follow up these erroneous conceptions in detail, 
but it would lead us too far afield for the purposes of the 
present discussion, which I may be content to close with 
the following remarks. If a and b are as we have thus far 
been regarding them, not things belonging to a reality out- 
side and independent of our thought, but simply contents 
of possible ideas, like red and yellow, straight and curved, 
it will follow that a relation between them can exist only so 
far as we think it, and by the act of our thinking it. Only 
such is the constitution of our soul and such do we assume 
that of every other soul to be which inwardly resembles 
ours, that whenever and by whomsoever they may be 
thought, they must always produce for thought the same 
relation, a relation which has its being only in thought and 
by means of thought. This relation therefore is indepen- 
dent of the individual thinking subject, and independent of 
the several phases of his thought ; this is all that we mean 
when we regard it as subsisting in itself as between a and b, 
as an object having a permanent existence of its own, which 
our thought discovers. It has in fact this permanent and 
assured character, but only in the sense of being an occur- 
rence which will always repeat itself in our thinking in the 
same way under the same conditions. And this holds not 
only of difference but of every relation whatever which we 
may discover between a and b. Every time that any mind 
forms the idea of a perfect circle it will be found, in this 
case, it is true, only through a chain of intermediate ideas, 
that the ratio i : 7T obtains between the diameter and the 
circumference ; this proportion therefore is valid in itself ; 
but although thus possessing objective Validity, it possesses 
Being only in the form of the thought which apprehends it. 

The case is different if a and b are taken expressly to 



Chap. IV.] RELATIONS AND REACTIONS. 263 



signify realities, things, beings, which we do not create by 
thought but recognise as objects outside thought. In that 
case the name Relation expresses less than we have to sup- 
pose as really obtaining between the related things. Only 
so long as we are merely placing the thinkable contents of 
this a and b by a voluntary act side by side for comparison, 
would a proposition affirming a relation between a and b, or 
more properly in this case between the ideas or thought- 
pictures of them, adequately express our meaning. If on 
the other hand we are led in order to explain some con- 
nexion between these ideas which perception has thrust 
upon us, to have recourse to a relation C such as to subsist 
not between the ideas but between the things a and b them- 
selves, of which the ideas are the thought-pictures, then we 
must recognise that this C which we have invoked cannot 
be a relation between a and b, cannot any longer therefore be 
a relation in the ordinary sense of the word at all. For it is 
thought and thought only which, passing from the idea a to 
the idea b, and becoming conscious of the transition, creates 
that which we call here a ' between/ and presents it as a 
mental picture which thought finds intelligible ; accordingly 
it must always be a vain endeavour to attempt to ascribe to 
this relation, which at once separates a and b and brings 
them together, and which is nothing more than the recol- 
lection of an act of thought performable only by the unity 
of our consciousness, — to ascribe, I say, to this relation a 
real validity in the sense of being something in itself apart 
from the consciousness which thinks it. This supposed 
' relation ' can only subsist independently of our conscious- 
ness, or objectively, if it is something more than relation, 
and then it subsists not betzveen a and b (for this ' between ' 
has no existence except in us), but rather in them, as an 
influence which they reciprocally exert upon and receive 
from each other. It is merely for us when we think it that 
such influence takes logical shape in the weakened form of 
a relation, which no longer expresses its full significance. 



2 64 REAL AND FORMAL VALUE OF THOUGHT, [Book III. 

I must leave it to the Metaphysic 1 to show what are the 
conclusions to which this observation leads ; to certain 
questions nearly connected with it I shall return directly. 

339. The comparison of a and b does not lead merely 
to the affirmation of identity or difference ; we also try to 
present identity in difference under the form of a universal 
as the content of a separate idea C. It is a criticism fre- 
quently made in Logic that our general conceptions do not 
possess the fixity with which ordinary thought credits them j 
their content is formed and their structure developed little 
by little, and the same conception means different things at 
different stages in the evolution of our growing knowledge, 
as fresh experiences continually enrich it. This is very 
evidently true of those conceptions whose content is drawn 
purely from experience, and therefore can only become 
gradually known to us ; on the other hand the conceptions 
of an integer or a fraction, a line or a figure, will not be 
found necessarily imperfect in the same way. The concep- 
tion of a triangle as such contains in it no more to the 
geometrician than it does to the scholar who follows him ; 
the difference is that to the geometrician it suggests nume- 
rous relations which the scholar is as yet unacquainted 
with, and in this way the conception of the triangle as such 
appears to be richer in content to the one than it is to the 
other, whereas the truth is that it is only his knowledge 
about it which is more extensive. But leaving this, the 
point I wish to emphasise is that a general conception, even 
if we consider merely its content at any one moment, indi- 
cates a task which no actual idea, that can be presented to 
the mind, can fulfil. A specific red or blue colour we can 
see, colour in general can neither be seen nor yet presented 
in the same sort of imaginative embodiment as the images 
of red and green recalled in memory. He who speaks of 
colour in general reckons on his hearer first of all summon- 
ing up the mental picture of some definite colour, red 
1 [Cp. 1 Metaphysic/ § 8o, and Book III. ch. 3.] 



Chap. IV.] HOW UNI VERSALS ARE THOUGHT, 265 



perhaps, which however is accompanied at the same time 
by a negation by which it is made to stand not for itself, 
but as an example of colour in general. To this negation 
however, if it is not to deprive the idea of all content what- 
soever, he can only give effect by calling up at the same 
time the ideas of other definite colours to his imagination, 
and becoming aware in passing from one to the other of the 
common element which remains constant throughout the 
changes of his conscious states. 

It is a series of psychical operations of this nature which 
is the task prescribed to us, when we hear the name of any 
universal ; but that towards which those operations are 
directed can never be presented as an actual idea ; we can 
never separate that which makes red and green colours 
from that which makes red red and green green. It is 
commonly admitted as a self-evident truth, that the class to 
which a real object belongs is not itself real ; this individual 
horse we see, horse in general is nowhere to be found ; but 
it has to be understood that in thought too the universal is 
never more than an idea strained after but always unrealised, 
floating over the forms of the individual instances of it 
which are imaged in the mind. To these purely inward 
operations of thought no objective significance can attach ; 
they remain subjective efforts of our mind, and the very 
form in which we express the result to which they lead us, 
' in intension the universal is included in the particular, in 
extension the particular in the universal,' merely indicates 
in the symbolism of space those operations of thought 
through which the mind endeavours to represent as an idea 
the objective relation between them. And now inasmuch 
as, more than this, we never find the object of our search, 
our universal, in actual presentation at all, we are led to ask 
whether it really has any objective significance ? Or are we 
to approve an opinion widely current, that it is merely the 
mechanism of the mind which misleads us into grouping 
similar impressions under general names by blurring the 



2 66 REAL AND FORMAL VALUE OF THOUGHT. [Book III. 

real distinctions between them to the prejudice of accurate 
thought ? This theory however in fact acknowledges that 
which it sets itself to deny. In order to make the fact in- 
telligible that not all but only similar ideas are thus drawn 
together under a common name it presupposes the fact of 
that similarity, and clearly, with it, only in another form, 
the objective validity of our assumption of a universal, 
which, however inseparably, is contained in them. On the 
other hand, if we could merely point to an innate tendency 
of thought to search for a universal, such tendency might 
very well be without any objective significance, but the fact 
that the object of the search is found gives it such a signifi- 
cance at once. 

This is only an apparent contradiction to what I was 
saying just now, for although the universal cannot be held 
before the imagination, the effort to think it is still not 
without result. We could not so much as bring red and 
blue under the general name of colour, did not that com- 
mon element exist in them, to our consciousness of which 
we testify in framing the name ; we could form no class 
notions of animals and plants if the marks of individual 
plants or animals, and the modes in which those marks are 
conjoined, did not really possess such points of comparison 
as allow us to arrange them under general marks and forms, 
and thus by setting these in the place of the merely in- 
dividual, to construct the thought-form of the class, however 
impossible it may be to picture it to the mind. Thus in the 
fact that we are able to think a universal, there is undoubt- 
edly contained a truth of real and objective validity; the 
contents of the world of ideas which thought does not 
create but finds do not fall into mere individual and atomic 
elements, each one admitting of no comparison with the 
other, but on the contrary resemblances, affinities, and re- 
lations exist between them, in such wise that thought as it 
constructs its universals and subordinates and co-ordinates 
the particulars under them, comes through these purely 



Chap. IV.] NOMINALISM AND REALISM. 267 

formal and subjective operations, to coincide with the nature 
of that objective world. 

340. If we pass from these more simple instances to the 
main forms of logical thought, and enquire into the signifi- 
cance of universal notions, we are met by the controversy 
between Nominalism and Realism, which excited such 
passionate agitation in the middle ages. To both parties 
the question at issue had other than a purely logical im- 
portance ; the metaphysical interest predominated, leading 
them to think of the world of ideas mainly in its relation to 
the world of things. Thus Realism first misunderstanding 
and then exaggerating the independence of the Platonic 
Ideas, came to look upon the general notion as the only 
real existence in things, all distinctively individual character- 
istics being relegated to the position of merely transient and 
subordinate though mysterious appendages to the eternal 
substance of the universal. Nominalism starting from the 
sound Aristotelian doctrine, that reality of Existence belongs 
only to the individual thing, found no way of reconciling 
this with the Validity of the universal. Thus the Nominalist 
came to regard general notions as at the best mere aids to 
the mind in the arrangement of its ideas, possessing no 
significance whatever in relation to the things which the 
ideas represent. They even erred so far as to deny them so 
much value as this, and to declare them to be mere sounds 
which may be uttered and heard, but are wholly devoid of 
content or meaning. 

I am desirous in the first place to avoid dealing with the 
subject thus exclusively in relation to the question of ex- 
istence, which involves an undue limitation of the issue. 
In Mathematics where we find ourselves dealing not with 
existing things and their essence at all, in Moral Philosophy 
and Jurisprudence where we speak of virtues and crimes, 
which ought or ought not to exist, more than this, when in 
actual life we endeavour to arrive at a decision in a matter 
of importance by bringing the given case under a general 



2 68 REAL AND FORMAL VALUE OF THOUGHT. [Book III. 

notion : — in all these instances we meet with the universal 
and its laws, in dealing with objects which are given us as 
matter of knowledge although they are not things \ 

341. If we can get out of the habit of always thinking 
exclusively of class notions in natural history as examples of 
the universal, if we recollect that we also frame general 
notions of figures and numbers, events and relations, truths 
and errors, the wild ambition to ascribe to general notions 
as such a reality like that of things, or at any rate of some 
actual existences, vanishes at once. To the original forms 
of substantial existences, of the plant, the animal, the 
human being, our imagination may if it pleases attribute an 
independent and eternal existence in a hypostasized world 
of ideas, as objects of intuition to souls which are yet 
unfettered by the limitations of an earthly existence. But 
the general notions of rest and motion, resemblance and 
contrariety, activity and passivity, could not possibly exist 
side by side with the former even in a world of ideas, they 
could only possess validity as predicates of the ideas. This 
fact, from which it is easy to see that there is no escape, we 
do indeed sometimes forget. We are tempted to treat 
qualities, relations, or occurrences, to which some prominent 
interest attaches, objects of our reverence or of our dread, 
as universals with a reality like that of an actual existence, 
misapprehending their purely predicative nature. We speak 
for instance of ' the beautiful ' as of a being which is merely 
to us unapproachable, but in itself an object of possible 
intuition ; we speak of 1 sin ' not merely as of an act which 
becomes real when we commit it, but as if it were a sub- 
stantial force which operated upon us with an independent 
reality of its own. We confound the importance which 
belongs to the content of such conceptions in the entire 
system of the world with a form of reality which it cannot 
possess, and in attributing which to them we are merely 
expressing in the most emphatic terms at our command 
1 [' Sachlich, aber doch nicht dinghaft.'] 



Chap. IV.] VALIDITY IS NOT EXISTENCE. 269 

their independence of our recognition of them. This 
mistaken habit of thought, which is not altogether harmless, 
is however here easily renounced ; it is only from that class 
of general conceptions the nature of whose content necessi- 
tates its being apprehended from the first in a substantive 
form, that this hypostasizing tendency continues to draw 
support. Here too however it has to give way before a very 
simple consideration. We are not content to frame, starting 
from the particular of perception, a single universal Q, but 
we go on to combine this with others like it in a higher 
universal P, and as we proceed with this operation, it rests 
within wide limits with our own logical good pleasure to 
determine through how many such links in the chain of 
universals we may choose to connect our Q with the highest 
universal A, at which the process of abstraction will be 
arrested. Each one of these universals would have an 
equal right to such substantial existence ; side by side with 
animal in general would appear vertebrate in general, 
mammal in itself, one-toed animal in general, horse in itself, 
black horse in general, all equally real. I say deliberately 
side by side with each other, for in fact our imagination is 
totally unable to transfer that relation of subordination 
through which in our thought one such general notion 
includes another, to beings such as these, which are con- 
ceived as possessing all alike actual existence. Placed thus 
however side by side with one another they could no longer 
have the meaning which they purport to have. Thus we 
find ourselves confirmed in our conviction that this Reality 
which we desire to recognise in the general notions which 
are created by our thought is a reality which is wholly 
dissimilar to Existence, and which can only consist in 
what we have called Validity or in being predicable of the 
Existent. But how much of the full meaning of a general 
notion possesses this validity, and what is the meaning of 
possessing £ validity ' at all, are questions which need some 
further discussion. 



270 REAL AND FORMAL VALUE OF THOUGHT. [Book III. 

342. I remind my readers to begin with that we are not 
concerned with the question of the objective value which 
may attach to one or another of the general notions evolved 
by thought in virtue of its content being correctly con- 
structed ; the question relates to the general significance of 
the logical form as such. That this like every other of the 
forms which logic prescribes as ideals may be given a 
content which is not adapted to it, needs no special men- 
tion, but a critical review of the countless modes in which 
the form of the notion may be applied is not our business 
here. To proceed then : we saw that any content of 
thought S is conceived under the form of the notion, when 
we do not merely grasp its manifold constituents as some 
sort of whole, but present to ourselves at the same time a 
universal M whose general characteristics jP } Q . . . standing 
in determinate relations to one another, become severally 
modified and defined in S in the specific forms p% q*. This 
constitution of the logical notion does not correspond to 
anything which takes place in things or external objects 1 
themselves ; and neither does it answer to the actual nature 
of a content which is presented to us as matter of know- 
ledge but not as a substantial thing. There is no moment 
in the life of a plant in which it is merely plant in general 
or conifer in itself, awaiting some subsequent influences 
answering to the subsequent logical determinations in our 
thought, to settle the question what particular tree it is to 
grow up into. It is true that the plant is not while still in 
the germ its future self in perfect miniature ; still its manner 
of development is not that certain conditions superadded 
from without produce a special determination of character- 
istics which were present in a general and indeterminate 
shape ; on the contrary its characteristics are already fully 
determined when the conditions enter in. From the two in 
conjunction new results are produced of which it is mis- 

1 ['Eines Dinges oder Gegenstandes,' contrasted with 1 Inhalt ' (con- 
tent) and ' sachlich' (matter of knowledge) in the following clause.] 



Chap. IV.] THINGS ARE NEVER ABSTRACT, 2)1 

leading to say that they were contained in the earlier and 
more general properties as mere potential species and are 
now for the first time actualised to the exclusion of all other 
alternatives. An ellipse has no natural existence and de- 
velopment like a plant ; still here too it is not the only way 
of arriving at a true apprehension of its nature to think of it 
first as a curve possessing the general properties of all curved 
lines, and then to define those properties further till we 
reach the particular form of them which belongs to this par- 
ticular curve. We may indeed arrive at the conception of it 
in that way — supposing for instance an unpractised memory 
only allows us at first to recall the general outlines of the 
figure required, and we need subsequent reflexion to draw it 
exactly; but in the mathematical equations, whether they 
refer the shape of the line to arbitrary points of origin, 
or take account of some graphical method by which it may 
be generated, the curvature itself is not directly expressed at 
all ; it only appears as a consequence which may be deduced 
from the definite ratios of the co-ordinates. These con- 
siderations hold equally as concerns the subordination of 
notions to one another in classification; it has no real 
significance in relation to the actual structure and develop- 
ment of things themselves. This horse was not to begin 
with animal in general, then vertebrate in general, later on 
mammal, and only at the last stage of all horse ; nor can we 
by any means at any moment of its life separate off as an 
independent set of qualities the more fully defined group of 
properties which make it a horse, from the more general 
and less determinate which would make it a vertebrate, or 
from those most indeterminate of all which would merely 
constitute it an animal as such. Add to this that not only 
do different classifications of the same objects conflict owing 
to imperfect knowledge and observation, and thus introduce 
various and diverse ladders of universals between the highest 
universal and the objects, but the logical right of thought is 
incontestable to start from any point of view it pleases, and 



272 REAL AND FORMAL VALUE OF THOUGHT. [Book III. 



so to subsume the same object 5 under different general 
notions, or to construct its conception of the object by 
means of several widely divergent series of successive deter- 
minations. In such a case we are at liberty to ask with a 
view to the particular purpose of any enquiry, which of these 
various constructions is to be preferred, as presenting the 
object in the form in which it can be brought most con- 
veniently under the principles which happen on the particular 
occasion to be our guiding principles ; and if we knew 
ourselves to possess a knowledge of the supreme principles 
of the universe, such as would contain within them the key 
to all problems which could arise, then we might go on to 
select out of the various possible conceptions of an object 
that highest or best conception, which would indicate its 
place in this supreme classification, and in which all the 
other conceptions of it would be contained as logical con- 
sequences. Still greatly as the value of this conception for 
knowledge would be enhanced if this ideal were attained, 
from the importance which would then attach to its content 
and to the mode in which this content would be internally 
connected, for all this the Logical structure which belongs 
to it as a conception would still represent no Real structure 
corresponding to it in the object itself. 

This value for knowledge however, which we do not 
dispute, gives us the other side of the question, that which 
we mean when we all insist in spite of everything, that the 
general notion and that classification do at all events contain 
something which has to do with the thing itself. We shall 
perhaps be disposed to express it by saying that the whole 
series of intersubordinated universals are contained not actu 
but potentid in the essence of the thing itself ; and this pro- 
position will be extended to other and different ways in 
which a given content is constructed or conceived : not 
really but potentially is every mark of division contained in 
the continuous magnitude which we break up by means of 
it ; potentially all simple motion in a straight line contains 



Cha?. IV.] REAL VALUE OF THE JUDGMENT. 273 



in it the two component motions into which we may choose 
to resolve it ; 7 is not 4 + 3, but certainly it admits of those 
figures being substituted for it for purposes of calculation. 
We may interpret these phrases into more definite language ; 
all the processes which we go through in the framing of 
conceptions, in classification, in our logical constructions, 
are subjective movements of our thought and not processes 
which take place in things; but at the same time the nature 
of those things, of the given thinkable contents, is so con- 
stituted, that thought by surrendering itself to the logical 
laws of these movements of its own, finds itself at the end 
of its journey if pursued in obedience to those laws, co- 
inciding with the actual course of the things themselves. 
The paths however which it can pursue with equal prospect 
of success in passing from one element of its content to 
another, are many and not one ; in countless directions the 
world of possible ideas extends and is knit together, a 
diversely articulated system of coherent connected series, 
and thought when it moves from one member in the 
system to another, choosing its path at pleasure but always 
observing its own laws, resembles in some sense a melody 
whose course we cannot predict yet which strikes always 
definite intervals in the scale each with its determinate 
harmonic relations. 

343. When we come to the judgment we find that not 
only its logical form but its content for knowledge which is 
expressed through that form has in itself no direct Real 
significance. We give utterance to the categorical pro- 
positions ' This tree blossoms,' £ Atmospheric air is a 
permanent gas,' ' Every triangle has its angles equal to two 
right angles.' In the first case it is merely thanks to the 
subject-matter of the proposition that we are able to ascribe 
to the tree an existence which really is independent of the 
temporary condition of blossoming, that is to say that the 
subject and the predicate are actually related and separated 
as we divide and connect them in the form of the judgment. 

Logic, Vol. II. T 



274 REAL AND FORMAL VALUE OF THOUGHT. [Book ill. 

In the two other cases this separation is not to be found in 
the thing itself, it is a purely subjective movement of thought 
arbitrarily selecting one particular constituent in a whole 
which is really a unity, to be made the object of separate 
attention. The differences in the copula again in the three 
propositions are due merely to the imagination, which 
adapts itself to the peculiarities of each separate subject- 
matter, and finds an expression for them in language ; logic 
itself testifies by representing all judgments under the 
symbolic form £ is P, that in the uniform copula 6 is ' all 
objective distinctions in the connexion between £ and P 
are obliterated. They may be related as whole and part, as 
a thing to its transient states, or as cause to effect ; in the 
form of the judgment they appear solely as subject and 
predicate, two terms which denote merely the relative 
positions which the ideas of them assume in the subjective 
movement of our thought, and tell us nothing as to the 
objective relation in itself which if 'it becomes an object of 
thought compels the ideas to assume those positions in our 
thoughts. Once more, in Hypothetical judgments we do 
indeed appeal to an objective relation of this sort, but in the 
form of the judgment we neither express it nor make it 
intelligible. The conjunction of antecedent and consequent 
in the form, ' If B is true F is true,' in itself affirms no more 
than the proposition that B and F belong both together and 
in some way not defined to a single notion M. The fact 
that we notwithstanding divide this coherent unity and place 
one part of the notion in front of the other, so that by 
reason of the inseparable connexion between the two, the 
one becomes antecedent, the other consequent, — all this is 
once more simply one of those subjective movements of 
thought which do not take place in the content of the 
notion. And this subjective character of the movement 
is shown by the fact that we have it in our power to reverse 
its direction. We say, ' Every equilateral triangle is equi- 
angular,' or ' If a triangle is equilateral it is equiangular," 



Chap. IV.] ANTECEDENT AND CONSEQUENT. 275 

but we might say equally well ' If it is equiangular it is 
equilateral.' That which constitutes the objective content 
here is the undivided thought or the intuition of the equi- 
lateral and equiangular triangle ; the two constitutive 
elements, equality of sides and equality of angles, are 
simultaneously present in it, but thought taking an arbitrary 
starting-point at one or the other moves up and down 
between them dividing and uniting in its own fashion. 
This holds of all judgments which like those of mathe- 
matics are occupied with the ideal and not with the 
actual. They would all admit of simple conversion, if 
their expressions in language through the medium of pro- 
positions allowed of all the conceptions which occur in 
them being as precisely defined as is the case in the form 
of the equation. 

If on the other hand our hypothetical judgments relate 
to data of reality, in such cases our intention is certainly 
that the antecedent and the consequent are to be taken as 
not interchangeable, but the hypothetical form of the judg- 
ment does not in itself express the condition which makes 
that assumption true. For given the antecedent B there is 
logically no interval left which separates its validity from 
that of the consequent F; the two together constitute, in 
perfect accord with that which the hypothetical form of 
judgment itself affirms as its result, a single process M 
which can be expressed in a judgment. And further inas- 
much as if we take our conception accurately, leaving no- 
thing out and adding nothing to it, no F 1 can be connected 
with our B but F only, and no B l with F but B only, it 
follows that we pass in thought with equal right and neces- 
sity from either of the two starting-points taken at pleasure, 
to the other, from B to i^just as much as from Fto B ; we 
know the consequent from the antecedent and the antece- 
dent from the consequent. That in actual fact there is here 
some circumstance which makes B and B alone the Ante- 
cedens, and jPand F alone the Consequens, we are very well 

T 2 



276 REAL AND FORMAL VALUE OF THOUGHT, [Book III. 

aware, because we are acquainted with the subject-matter 
under consideration, but it receives no expression through 
the form of our logical act. For that form depends upon 
nothing more than the abstract notion that Fis in a general 
sense co?iditioned by B ; but this, a mere abstract relation, 
is as shown already, something less than anything that we 
obtain in reality between B and F as things or events. A 
relation through which B and B only is to be the antece- 
dent, and to be a real antecedent, can only actually obtain 
if B is cause and Reflect ; but in the hypothetical judgment 
instead of this real and specific relation of causality we have 
nothing but the vague and general relation of conditioning 
in the abstract, which thus has no significance for reality 
whatever. 

Finally Disjunctive judgments do not even purport to 
express any reality at all ; the process of wavering undecided 
between several mutually exclusive predicates can answer to 
no process in the real world ; it remains a state of our think- 
ing, to which the adequate data for the knowledge of reality 
are lacking. 

344. A brief consideration of the various forms of Syllo- 
gism leads us to similar results. We shall be most readily 
disposed to ascribe a Real significance to those Figures of 
Subsumption which arrive at their conclusions by bringing 
the particular under the universal, for this subordination we 
do certainly regard in the sense already sufficiently ex- 
plained as a notion which possesses an objective validity in 
relation to everything that can be presented to the mind as 
an idea. 

Still here also the logical form of the argument does not 
correspond to anything that takes place. In mathematical 
syllogisms the universal major premiss, from which we de- 
rive our more particular conclusion, has no priority of truth 
as compared with the conclusion or with the minor premiss ; 
all three are parts of one eternal truth, all possess a simul- 
taneous validity. The priority of greater simplicity or more 



Chap. IV.] SUBJECTIVE VALUE OF SUBSUMPTION. 277 



immediate evidence the major premiss may indeed possess, 
but both the one predicate and the other would belong to it 
in relation to our thought only, without giving it any supe- 
riority in itself over other propositions of equal certainty. 
Lastly there is nothing in the form of inference by Sub- 
sumption which obliges it to start from a major premiss of 
this simple character at all ; on the contrary the simul- 
taneity of the connexion which obtains between the entire 
body of mathematical truths allows the simpler among them 
to be derived as limiting cases from a logical connexion of 
less simple, no less than the other way, and always in this 
figure of Subsumption. 

This purely subjective significance of the form of the 
syllogism we sometimes forget in applying it to matters of 
fact. So long indeed as the universal major expresses a 
highly concrete and specific truth, when for instance we say 
'All animals breathe, 7 we never question that such a major 
premiss cannot designate any reality which is prior to the 
validity of the conclusion 6 Fishes breathe,' anywhere but in 
our thought. Yet when we turn to the most universal 
principles of the system of things, the impulse comes back 
upon us to give to the expression of those principles, the 
most universal laws of nature, which present themselves as 
major premises in our enquiries into the order of the world, 
a real priority, which is in fact wholly inconceivable, to the 
processes in which they are to hold good. This impulse is 
not without danger to the soundness and consistency of our 
metaphysical theories ; it leads to a superstition which has 
far reaching consequences, that the reality oftfie world may 
be derived from something which is unreal and which is yet, 
esse ntial and possessed of a regul ativ e j^wer^whereas on_ 1 
the contrary we have thoroughly to convince ourselves that^ 
all necessary truths, to w hich we imagine that we can sub^ 
ordinate the existent as if it were something merely secon- 
dary and additional, are simply the nature and self-consistency,, 
of the existent itself, and are only disengaged from it by a 



2 78 REAL AND FORMAL VALUE OF THOUGHT. [Book III. 



reflective act of thought and credited accordingly with a_ 
prior and regulative character to which they have no claim. 

Inferences by Induction do not give occasion to this sort 
of misunderstanding ; no one fails to see that the synthesis 
of particular facts in a general, not merely a universal 1 , pro- 
position is not the real ground of the validity of the general 
proposition but only of our apprehension of that validity. 
Still more convincingly does the variety of forms, which a 
Proof may assume, witness to the merely subjective signifi- 
cance of the several inferences of which it is made up. 
How many different proofs, direct and indirect, progressive 
and retrogressive, all equally adequate, may be given for 
one and the same proposition ! How many even in the 
form of direct progressive argument alone ! And supposing 
that in fact one out of the many could possess the preroga- 
tive of alone exhibiting the essence and actual structure of 
the thing, still the mere fact that other forms of proof are 
possible would always show that it is not the logical Form 
by itself which occasions or expresses the Real validity of 
this particular form of proof, but that its superiority over 
other forms of proof lies in the content which we have 
taken and conjoined in this form. Lastly in regard to the 
final operations of thought with the account of which the 
doctrine of pure Logic concluded, we saw there that Logic 
does in those operations strive to discover some Forms in 
which the proper essence of the thing, as distinguished 
from our mere subjective and haphazard notions about it, 
may be exhibited. But there too we come to the conclu- 
sion that those Forms turn out to be far wider than that 
which they purport to contain. If the proper essence of 

/"the thing does make its way into our thought, it can only 
be apprehended under these Forms, but the Forms do not 

! create it and do not fully express it ; they admit always of 
fresh applications which issue as we are ourselves conscious 
in merely subjective notions, and from among which the 
1 [See § 68, sup.] 



Chap. IV.] ANTITHESES ABOUT THOUGHT 279 

selection of the more trustworthy in relation to reality can-i 
not be made by the help of Logic but only through know- 
ledge of the subject-matter, if such knowledge is forth- 
coming. 

345. It is now time to determine more exactly the mean- 
ing of certain expressions in the use of which I have hitherto 
been somewhat less precise. We have spoken of Subjective 
and Objective, of Formal and Material \ of Formal and 
Real significance, as applied to the Forms of Thought. 
The three pairs of antitheses do not coincide. If we dis- 
tinguish, as we have done, between the logical act of think- 
ing, and the thought which it creates as its product, the 
former can claim only a Subjective significance ; it is purely 
and simply an inner movement of our own minds, which is 
made necessary to us by reason of the constitution of our 
nature and of our place in the world, and through which we 
make that Thought, for instance the distinction which exists 
between a and b, or the universal C which is contained in 
them both, an object for our own consciousness. In the 
same way every one who desires to enjoy the prospect 
from a hilltop has to traverse some particular straight or 
winding path from the point at which he starts up to the 
summit which discloses the view; this path itself is not part 
of the view which he wishes to obtain. The Thought itself 
on the other hand in which the process of thinking issues, 
the prospect obtained, has Objective validity; the various' 
paths followed by various travellers once traversed and left 
behind, the scene which opens before them is the same to 
all alike, an object independent of the subjectivity of the 
individual ; it is not merely one more affection of his con- 
sciousness which he experiences, but an object presented to 
his thought which also presents itself as the same self- 
identical object to the consciousness of others. 

The second antithesis 2 throws light on the same state of 
facts from another side. It would not be sufficient to call 

1 [' Sachlich/ opp. to ' formal.'] 2 [' Formal' and ' sachlich.'] 



28o REAL AND FORMAL VALUE OF THOUGHT. [Book III. 

the operations of our thought Subjective and nothing more. 
The term would simply separate them from that which ac- 
tually goes on in the object-matter 1 with which they deal, 
leaving it quite obscure what the relation is in which they 
stand to it ; yet after all some such relation there must be, 
if the Logical Thought in which they issue, is to possess an 
Objective validity which does not belong to the thinking act 
which issues in it. Accordingly we call the logical opera- 
tions not Subjective merely but Formal because their 
characteristics though not the actual determinations of the 
matter they deal with 2 , yet on the other hand are Forms of 
procedure the very purpose of which is to apprehend the 
nature of that subject-matter, and which therefore cannot 
stand altogether out of connexion with that which there has 
place. 

Upon this point the illustrations adduced above will re- 
move all uncertainty. The limitation to a merely Formal 
validity showed itself in the fact that there may be several 
processes of thought equally successful in view of the result 
arrived at, all, that is, leading to the same final thought- 
product, or the same material result. No one of them 
therefore can have an exclusive significance as regards that 
determinate matter and content with which all are equally 
concerned; all alike are merely forms of procedure, em- 
ployed to reach a certain result which once obtained is valid 
independently of the path which led to it. But clearly it 
would be impossible to arrive by all these different paths at 
the summit from which this prospect opens, if they were 
not all included with their determinate positions and rela- 
tions the one to the other within that same geographical 
territory, the remaining part of which is what constitutes 
the landscape which is commanded from the summit. 
Herein consists the positive element which this second 
antithesis affirms of the processes of thought 3 ; each is one 

1 [' Von clem Verhalten der Sachen.'! 2 [' Der Sachen.'] 

3 [They are not merely ' subjective ' but also 1 formal.'] 



Chap. IV.] 



INDEPENDENT REALITY. 



281 



among the various ways in which the variously ramifying 
systems of the world of fact make it possible for us, by 
reason of its universal interconnexion, to arrive by a pro- 
cess of movement from point to point within that world, at 
a determinate objective relation, although the particular 
movement chosen neither is nor yet copies the way in 
which this relation itself arose or now obtains. 

The third antithesis 1 is not merely another way of ex- 
pressing the second ; it relates to a specific question. We 
regard every content of thought as having a material value 2 
if it has a fixed Objective significance in the sense above 
explained — ideas of the non-existent no less than of the 
existent ; by the term ReaP we should have to understand 
only things and events in so far as they exist and occur in 
an actual world of their own beyond thought. Now it is 
out of the question that this kind of Reality should move 
and have its being in the forms of the Concept, of the 
Judgment or of the Syllogism, which our thought assumes 
in its own subjective efforts towards the knowledge of that 
reality. But even the logical thoughts which are the issue 
of those operations have not in relation to Reality in this 
sense the immediate and material validity which belonged 
to them in relation to every content of thought as such. It 
will be better to reserve for the Metaphysic 4 the fuller dis- 
cussion of this important point ; a reference to the illustra- 
tions already adduced will suffice in the way of a preliminary 
elucidation. 

We saw that the notion of a condition is inadequate to 
denote that which we mean by a relation which subsists in 
actual fact between two real elements ; so to subsist, it 
would have to be more than a relation, it would have to be 
nothing less than interaction. This being so, it was in that 
Real connexion between the Real elements that the cause 
resided which brought their phenomenal appearances for us 

1 [In German 1 Formal ' and ' Real.'] 2 [' Sachlich gegeben.'] 

3 [German 1 Real.'] 4 [See * Metaphysic,' § 81.] 



282 REAL AND FORMAL VALUE OF THOUGHT. 

into that particular formal relation which we now, employ- 
ing a merely logical term, call a conditioning of one by 
the other. 

The same is true of all logical Forms. No real S can be 
subject and nothing more to a real P, which is its predicate 
and nothing more ; in actual fact P can only attach to 5 
either as a state which it passes through, or as an influence 
which it exerts, or finally as a permanent quality which 
belongs to it in the sense (a sense it is true at present some- 
what obscure) in which we contrast the metaphysical notion 
of a Quality with the merely logical notion of a Mark. It 
is not till one of these three relations has been affirmed that 
we understand what the meaning is realiter of the logical 
conception of S as subject and P as predicate. It is not 
till then that we have an actual state of things answering to 
the logical copula, which in itself leaves it quite undeter- 
mined what precisely we are affirming to have occurred to 
the real things in question, when we feel ourselves necessi- 
tated thus to connect the ideas of them. When then we 
employ such expressions as unity, multiplicity, equality, 
contrariety, relation, condition, so long as we use those 
terms by themselves, we have said absolutely nothing about 
the existent. We have still to show how it is brought about 
that the unity of the One is proved to be an actual reality, 
not merely a barren logical title ; how it is that what are 
many but identical, although in thought they simply are 
identical, nevertheless in real existence break up and be- 
come many; what is the one kind of reciprocal influence in 
which the opposition, what is the other kind of reciprocal 
influence in which the relation, of different existing things, 
shows itself to be real. 



CHAPTER V. 



The a priori truths. 

346. Let us put together once more the conclusions to 
which we have been brought. Neither in the content of 
our ideas nor yet in the reality which we regard as its source 
outside, was there anything to correspond to the logical 
processes of thought, which choosing their path at will, 
connected or separated the several constituent elements of 
which that content was composed. On the other hand, at 
least in relation to this content, without regard to that 
reality which may be its cause in the world outside, the 
Thought-product, in which it was the aim of the Thought- 
processes to issue, had, we saw, an objective significance. 
The differences, the resemblances, the contrasts, the sub- 
ordinations, of which we could only possess ourselves in 
consciousness by help of the discursive activity of Thought, 
passing backwards and forwards from point to point, had 
we saw an actual validity as applied to the apprehended 
content, although the content itself in no way participated 
in such movements. They subsist, as we saw, indepen- 
dently and objectively in the sense in which any other 
relation may subsist between the terms related. Real 
existence, that is to say, they can never claim except at the 
moments in which they are thought ; but on the other 
hand, such is the common constitution of all minds, that 
whenever the given terms of the relation a and b are 
thought, one and the same judgment C affirming this 



284 



THE A PRIORI TRUTHS. 



[Book III. 



relation between them is immediately and invariably pro- 
nounced. 

We are here brought back to the Platonic world of 
Ideas. All contents of possible ideas stand in fixed and 
unalterable relations, and by whatever processes or move- 
ments of thought, as our own pleasure or as chance 
determines, we may carry our attention from one to 
another, or in whatever order they may be one after 
another brought to our perception by occasioning causes 
even unknown to ourselves, — we shall invariably find the 
same relations obtaining amongst them which are given us 
once for all in the objective and endlessly complex structure 
of the world of Ideas. — So often as this proposition is 
insisted on it will be regarded as an entirely superfluous 
affirmation of that which is perfectly self-evident, and just 
as often I must repeat that the very existence of this self- 
evidence is the most astonishing thing in the world. 
Although an indispensable foundation of all thought, and 
just on that account passed over by us in our presumption 
as a mere matter of course, it is not even, as I observed 
before, a necessity of thought in the sense in which that 
character may be claimed for the particular relations which 
it includes within it. We cannot indeed fully realise in 
thought what the state of things would be if this fact were 
wanting, but still we can imagine a world in which it did 
not obtain; in which countless contents presented them- 
selves for our minds to form ideas of, but each one stand- 
ing in no relation to the rest, all so entirely disparate in 
nature that no two of them could be combined as allied 
species under any common universal, nor any two of them 
be pronounced to differ from each other more or less or 
otherwise than any other two. One postulate alone, in 
such case, Thought would be in a position to make, in 
obedience to its own law of Identity, namely, that each 
one of the contents must be identical with itself. This 
postulate would be the condition of their being presented 



Chap, v.] THOUGHT AND THE COURSE OF THINGS. 285 

to thought at all, and it might be fulfilled by such a world 
as I am supposing. But beyond this we cannot go. 
Thought may wish, in order to the possibility of its further 
operations, but it cannot demand as a necessity of thought, 
that between the different objects there should be found 
that graduated scale of affinities which alone enables it to 
accomplish the ends after which it strives, — it is not a 
necessity of thought that thought itself should be possible. 
And even supposing that by its own intrinsic power it 
could postulate those affinities, still it could not make them; 
it would always have to trust to their being given it by the 
grace of facts, ordered and arranged on principles which it 
could never have itself contrived, as series of tones or 
colours, or as differences in degree among things qualita- 
tively the same, or in any other way. 

But strange and important as is the fact that such 
affinities in the world of experience are actually found, it is 
not in this fact or in the consequences which follow from 
it, that the final goal of our enquiries lies. All that it 
guarantees us is the security with which thought is able to 
move within the world of ideas as such, to investigate the 
systematic and invariable connexions obtaining among the 
elements of that world, and by conjoining them one with 
another to construct new forms which will be found without 
fail in another and a predictable place in the world of ideas, 
so connected, finally, one and all together in various 
directions and at fixed distances, that the most diverse and 
the most roundabout tracks of thought may lead to the 
certain discovery of any one of them. This however by 
itself is not all that we are concerned to know. What we 
want to arrive at is the significance which is to be attached 
to this systematic arrangement of the world of knowledge 
in relation to that empirical and unsystematic order of 
events, in which a causal reality independent of thought 
presents contents of possible ideas to our perception. What 
we wish to understand is not only the classification of 



286 



THE A PRIORI TRUTHS. 



[Book III. 



things which is eternal, but also the course of things which 
is in change. 

347. The two are completely distinct. Perception does 
not present those objects to us in connexion which stand 
side by side as akin in the system of knowledge, nor is its 
entire history a periodically recurring procession of orders, 
genera, and species, following one upon the other in a 
descending scale as they do in the order of classification. 
Contemporaneous in different points in space, succeeding 
one another at different points of time, we find the most 
heterogeneous elements of that realm of contents phe- 
nomenally connected ; if laws in this scene of change there 
are, they are of a different kind altogether from the logical 
laws which have hitherto been engaging our attention. If 
we agree henceforth to designate the empirical course of 
phenomena as it is thus presented to us, Actual Reality 1 , 
then the question is as to the significance which our thought 
can claim in relation to it, since its affirmations even though 
retaining their validity, seem nevertheless to be incapable 
of controlling the order of connexion which the reality 
presents. For even supposing it to be true that a and b 
will exhibit, when given in actual perception, the same 
distinction and the same affinity which belong to them in 
our thought, still this tells us nothing as to whether they 
will actually be found in conjunction in perception, or 
whether that conjunction may not be an impossibility. 
Admitting the law of identity to hold without exception, 
still it does not profess to do more than affirm that now and 
always every a = a, and every b — b, tuhenever and wherever 
they may be found. But here at once the last clause is no 
longer part of the law of identity itself ; we append it 
because we know on other grounds that possible objects of 
thought are susceptible, over and above their eternal validity 
in the world of ideas, of an alternation of temporal reality 
and unreality in the world of phenomena. Of this the law 
1 [< Reale Wirklichkeit.'] 



Chap. V.] 



SUBJECTS AND CONDITIONS. 



287 



in question contains no indication, and cannot therefore in 
the least determine the order in which in that world 
whether in the way of simultaneity or succession, the two 
phenomena necessarily introduce or necessarily exclude one 
another. Again, the classifications by which we range our 
conceptions one under the other will be valid equally of our 
perceptions and of the timeless content of our ideas ; but 
when we bring a perceived object £ under the general con- 
ception M, although all the higher universal conceptions 
NL K which are contained in M are now valid of S too, 
still this deduction gives us no new objective knowledge, 
but only a logical analysis of what was already implied in 
bringing *S under M y — correct if this was correct, incorrect 
if the contrary, but in neither case enabling us to combine 
the S given in the perception with a P which has not been 
so given. 

Hypothetical judgments seem better adapted to an 
extension of knowledge. In so far as they apply to a 
subject S 2l condition x, and derive from the two together 
a predicate P, which was not already contained either in S 
or in x by themselves, they make at least a formal approach 
to that which we conceive to take place in reality. In the 
problematic antecedent they express the connexion of S 
and ^ as a possibility, and accordingly distinguish the 
thought content of it from the realisation which may be in 
store for that content in the actual course of events, and as 
to which they abstain from affirming anything. On the 
other hand, that condition once given, they do seem to 
anticipate the after perception and to define the new result 
which will necessarily follow in this perception. But now 
what is it that justifies us in subjoining to, or equating with 
a determinate S+x a determinate P? In thought it can 
rest only on this, that by means of a logical determination 
x we transform the notion S, which previously did not 
contain P, in such wise that now it does contain it ; and 
now it is of this new subject, not of the one we began with, 



288 



THE A PRIORI TRUTHS. 



[Book III. 



that we affirm the predicate P, which in fact we have 
already taken into it. But that which is directly presented 
to us in perception is something different from this. 
When in actual perception a new phenomenon x enters into 
relation to a previous phenomenon S, what happens as a 
rule is not that from the conjunction of the two in thought 
there results the subject S+x, from which thereupon the 
resultant phenomenon P would follow as a matter of course 
as if they were equivalent expressions. On the contrary 
the question has still to be solved, how it is possible for x 
so to transform S, that there may spring from it the con- 
ditions for the realisation of P which were before want- 
ing. Thus, wherever we apply hypothetical judgments to 
questions of the real world, they are always found to rest in 
the last resort upon certain presuppositions. They always 
assume the validity of certain propositions affirming the 
connexion of a particular condition with a particular conse- 
quence — a connexion which cannot be deduced from con- 
ceptions—to be a universal fact. If it is really universal, 
then thought can draw it out into its particular instances by 
a purely analytic procedure, but its real content appears, to 
begin with, as a synthetical judgment, which binds together 
as subject and predicate two conceptions, the contents of 
which mere logical analysis can never prove to be identical. 

848. Our hope then of mastering by thought the course 
of events in the real world, rests on three points. First, to 
no single constituent b of the ideal world can thought 
ascribe, over and above the eternal validity which within 
that world belongs to it, a necessity of realisation in the 
order of events in time ; it is only if this reality belongs as 
a matter of fact to a second such element a, with which b 
stands in necessary connexion, that it can then pass over to 
b also. All our knowledge therefore is in this respect 
hypothetical ; it strikes in at a particular point in a reality 
which it finds as a matter of fact given to it, in order to 
deduce from this real premiss as themselves real the con- 



Chap. V.] THE 1 ONTOLOGICAU ARGUMENT. 289 



sequences which attached to the thought premiss as 
necessary \ but it is never possible, starting from mere 
conceptions of thought, to prove the actual reality of that 
which is contained in them. And in fact the attempt has 
never been ventured upon except in the single instance of 
the ontological argument for the being of God. The 
temptation in this case was very intelligible. The con- 
ception of God as a necessary consequence b following 
from a reality a other than Himself, and given in percep- 
tion, contradicted our necessary idea of Him, for this very 
idea demanded that He should be conceived as the ground 
of all consequences. Hence, it seemed, nothing remained 
but to seek the reality of God in the idea itself of Him. 
True all that could really be found was the claim to reality 
which the idea carried with it. Beyond question the idea 
of God includes the idea of Being, and more than this, the 
idea of living Being ; for all other predicates by which we 
think of God as God, can only be unified, or even thought, 
when they are conceived as belonging to a real Being who 
fills time, and is capable of undergoing a change of states. 
But in this sense the idea of any being whatever includes 
the thought of that particular kind of reality which the 
nature and the mode of combination of its content require. 
The very notion of an organism is unthinkable without this 
assumption ; the properties of nutrition, growth, propagation 
of its kind, have no meaning when applied to a subject 
which does not exist, and just as little when applied to one 
which exists merely and has no faculty of development. 
If therefore the objects of our conception are to have 
reality at all, they must have that kind of reality which 
answers to their nature, Beings that of existence not of 
occurrence, Events that of occurrence not of existence, 
Relations neither the one nor the other, but a reality which 
consists in being valid of reality. It was a mere illusion to 
suppose that the case was different with the idea of God, 
and that it was allowable to look upon that notion of the 

Logic, Vol. II. U 



290 



THE A PRIORI TRUTHS. 



[Book III. 



highest reality which is necessarily included in that idea 
as equivalent to the reality of the whole content which 
included it. 

A class of arguments nearly allied to this, which pass 
from the incontestable value of an object of thought to the 
belief in its reality, have an appearance of committing a 
fallacy of the same kind, but in this case it is an appearance 
only. It is not altogether just to maintain that we believe 
in a supreme Good, in a life beyond the earth, in eternal 
blessedness, merely because we desire them. In reality 
such beliefs rest upon an extremely broad, though an un- 
analysed foundation of perception. They start from the 
fact of this actual world as it is given us in experience, in 
which we find certain intolerable contradictions threatening 
us if we refuse to acknowledge that these ways in which the 
structure of the world extends beyond our perception are 
real complements of that which we perceive. In form, 
therefore, this class of inferences is quite legitimate ; starting 
from the reality of a as given in experience, they connect 
with it the reality of o which is not so given, but which 
appears to follow from a as a necessity of thought. 

349. The second point alluded to is tacitly assumed in 
every argument, but seldom explicitly acknowledged as a 
necessary logical assumption. Clearly we could never hope 
to work upon reality through the medium of thought, if we 
were not in a position to assume in the empirical order of 
things the presence of universal law, which alone makes it 
possible for us to turn the formal laws of our thought to 
positive use. We saw that the real causes which determine 
the succession of our perceptions of possible contents of 
rJ thought are wholly independent of the systematic relations 
which we find between those contents when regarded as 
objects of thought simply. Whence then do we derive our 
assurance that there are reasons of universal validity at all 
determining this order of succession, and that the unknown 
cause of the experienced series of our perceptions is not 



Chap. V.] GROUND OF THE BELIEF IN LAW. 291 



simply playing with the elements of our Ideal world and its 
systematic classifications, itself void of all principle, bringing 
before us like a self-acting kaleidoscope now one arrange- 
ment of the picture and now another, but observing no law 
or order in its combinations ? 

We have no ground whatever for representing the wild 
disorder which this supposition implies as unimaginable ; 
there is a very great deal in the empirical world which 
we do not yet understand, that actually does still so appear 
to us ; if throughout the world of reality all regular law 
and regular relations were altogether absent, all we can 
say is that the same spectacle would then be presented 
to us everywhere which meets us now in cases where 
the laws are concealed from us. The laws of our thought 
would still hold good, but in the sense of an empty 
postulate, to which reality would offer no counterpart, 
just as there are many events even now to which we 
seek in vain to apply them, events which seem with their 
like conditions and unlike results to mock at our principle 
of identity. Nevertheless this assumption of an inde- 
pendence of law in the real world is maintained by no 
one ; in every case where observed phenomena might seem 
to force it upon us, we regard the state of facts so presented 
as simply a problem which awaits solution, and we never 
doubt that a wider experience will furnish links of con- 
nexion hitherto unobserved to restore order and regularity 
to the observed parts in which at present they are not 
to be found. 

Now on what does this confidence rest? The uni- 
versality of laws in the real world is neither in itself a 
necessity of thought, nor can it be deduced as a necessary 
consequence from given facts. We might have the right 
to say that the laws of space, even supposing space to 
exist only as an innate intuition in us, still must of ne- 
cessity hold good of all objects of our experience, for 
nothing will ever make its way into experience without 

u 2 



292 



THE A PRIORI TRUTHS. 



[Book III. 



having been already moulded in that form of space through 
which alone it becomes an object for us at all. But we 
cannot attempt to prove in the same way that unless there 
was a connexion according to law in the real world the 
experience which we possess would be impossible. That 
which we actually possess is merely a succession of ideas ; 
that this succession constitutes an unbroken connexion 
in accordance with universal laws, that is to say that 
experience in this heightened sense, as distinguished from 
mere perception, is also actually given us, — to affirm this 
is to confuse that which we know as a fact with pre- 
conceptions of our own which we bring to the facts. For 
r our actual knowledge amounts to no more than this, that 
a large number of occurrences admit of being regarded 
^ as if they were conditioned by universal laws ; there re- 
\ mains always a far larger number which we have not yet 
j succeeded in thus reducing to order. A reign of law 



I its rules, is therefore neither an actual nor a possible 
I outcome of experience, but only an assumption with which 
every enlargement of our experience is accompanied. 

We have therefore only two alternatives. Either we 
may acknowledge this assumption as an assumption and 
trust it, and thus credit ourselves with this one piece of 
certain knowledge, by the help of which our thought, 
crossing the boundary of its own domain, reaches one 
certain result as to the nature of reality; or we may look 
upon it equally as a mere assumption, and on that account 
distrust it, accepting thankfully such instances as confirm 
it, but always bearing in mind the possibility of finding 
1 ourselves stumbling at any moment upon ground where 
it no longer holds good. Whenever human reflexion has 
reached the point of a scientific view of the external world, 
it has without exception preferred the first of these two 
/ alternatives. Even those who are most careful to resist 
any undue encroachments of reason, and pride themselves 




admitting of no exception to 



Chap. V.] FREEDOM OF THE WILL. 



2 93 



on interrogating nature and nature only as to her own 
laws, never question the fact that such laws do universally 
obtain, they only insist that we know nothing about them. 
Only they do not observe, that in thus affirming the uni- 
versality of law they are passing beyond the data of reality, 
and are making in one clause an a priori assertion about it 
which the next declares to be illegitimate. 

The alternative theory may be thought to be discoverable 
in one particular instance, the belief in the freedom of 
the human will. As to the material rights and wrongs 
of this hypothesis, I am not here called upon to decide. 
But with regard to its form, it is only in appearance that 
it comes under the point of view in question. It does 
not assert that the same thing is free at one time and 
conditioned at another without any reason. On the con- 
trary, subjecting as it does one sphere of reality permanently 
and without exception to determination by fixed laws, 
and connecting the fact of freedom exclusively with the 
presence of a particular spiritual nature in the subject 
which wills, it does in fact assume that the system of the 
world is throughout a system of law, and merely ascribes 
to it the peculiar property of admitting at particular points 
in its course of the entrance of unconditioned elements, 
which once admitted into the world of reality thenceforth 
produce results which are conditioned by law. This theory 
also then, and more clearly still any theory which, denying 
freedom, brings the inner world as well as the outer 
under a system of determinate laws, permits itself in so 
doing to make an a priori affirmation concerning the real 
world, the universal validity of which experience as such 
can never prove. Whether it is justified in so doing, can 
never be decided by strict logical argument, for every 
attempt to prove this affirmation a necessity of thought, 
would leave the question of its validity as applied to the 
real world undecided. On the other hand, to attempt to 
exhibit it as agreeing with the nature of reality, would 



294 THE A PRIORI TRUTHS, [Book III. 

only be to repeat in a new form the old claim which it 
is desired to establish, the claim to be able to make 
a priori, that is to say universal- statements on the au- 
thority of thought alone, about that real world, of which 
experience can never give us universal knowledge. We 
have therefore the right to say that all our conclusions 
concerning the real world rest upon the immediate con- 
fidence or the faith which we repose in the universal 
validity of a certain postulate of thought, which oversteps 
the limits of the special world of thought. In point of 
fact this confidence which logic can never justify lies at 
the foundation of all logic, as it does also of that formula 
in which we described it as the universal tendency of 
thought to turn the observed fact of co-existence into 
coherent connexion. The methods of applied logic one 
and all have a meaning only on the assumption that that 
inward coherence and connectedness which this tendency 
ascribes to the real world does actually belong to it. To 
suppose it otherwise would be to cut away the logical 
standing-ground on which induction relies whenever it 
pronounces one inference drawn from experience to be 
even more probable than another; it would have to be 
content with rehearsing the premises, the conclusion would 
be wanting. 

350. There remains the third question. The assumption 
of a connected system of uniform laws embracing all reality 
does not by itself teach us what the particular laws are, 
in accordance with which a definite event b is conjoined 
with another event a. Further we have already satisfied 
ourselves that the mere analysis of the contents of the 
notions of a and b as such could never enable us to affirm 
that the realisation of the one must necessarily be followed 
by that of the other. Two courses remain open to us : 
either to lay claim to an immediate certainty of the uni- 
versal and necessary validity of synthetic judgments which 
nevertheless demand such a connexion, or else to content 



Chap. V.] EMPIRICISM PURE AND SIMPLE, 295 



ourselves with extracting all the particular laws of reality 
one by one from the evidence of experience by the help 
of the methods expounded in the last book. At this 
parting of the ways I wish by one general formula of 
ready worship to purchase a dispensation from any further 
glorification of the second of these two alternatives. It 
becomes in time wearisome to be told over and over 
again in endless iteration, how reason is to come to nature 
in a spirit of self-renunciation, how indeed from her own 
resources alone she cannot possibly decide a single question, 
and how she at once wanders off into a world of brain-spun 
phantasies if she does not at every step apply to experience 
for her data. Unhappily we cannot affirm that such 
warnings are superfluous, or that they are nowhere ap- 
plicable, for errors enough have been due to the neglect 
of them. Still any moral sermon becomes intolerable if 
it goes on for ever, and at last its only effect is that it 
moves us, as we are moved here, to ask the question 
whether the claims which the doctrine advocated holds 
up for our acceptance are not just as one-sided as con- 
fessedly those are which it undertakes to disprove. Can 
then, we ask, the purely empirical investigation of the 
laws of the actual world really solve its problem entirely 
from its own resources, calling in perhaps the aid of the 
law of identity, but otherwise without making assumption 
of any synthetic judgments a priori! That it cannot 
do this, was the doctrine of Kant ; if we arrive at a similar 
conclusion, we shall be championing a characteristic tenet 
of German philosophy, which has brought on us assaults 
from all nations. 

351. English scepticism in the person of Hume en- 
deavoured to restrict us on the one hand to the expression 
of mathematical truths, which appeared to Hume to rest 
simply upon the principle of identity, and on the other 
to the narration of the facts of history, which having once 
occurred are thenceforth matter of actual experience, and 



296 



THE A PRIORI TRUTHS. 



[Book III. 



can be expressed in synthetic judgments a posteriori. 
No scientific inference was possible, he thought, which 
should predict the occurrence of a b in the future on 
the strength of a given a which was not identical with it. 

Before I go on to discuss the last-named contention, 
it may be useful to point out, that if it is valid, then the 
previous contentions made with regard to mathematical 
and to historical truths cannot be. The possibility of 
synthetic judgments a posteriori is a point which does 
not sufficiently arouse our suspicions, because they are 
taken for simple expressions of experience, into which 
no admixture of too forward thought has made its way. 
But so long as they are judgments at all, no matter 
whether expressed in language or not, they are still not 
the facts given simply, but a preparation of the facts, 
made by reading into them an inner connexion which 
in immediate observation is not to be found. No narration 
of an event is possible except by combining together as 
subject and predicate one portion of the sensuous images 
which arose in us when we witnessed it, with another, 
and then going on to think in between the contents of 
these two conceptions a relation of action exerted on 
one side and received on the other, or again of mutual 
alteration of states, none of which relations are in the 
least degree given in the perceptions as such. 

It may be contended that the proposition Caesar crossed 
the Rubicon, means no more than that a certain partially 
changeable, but still coherent group of sensible impressions, 
which for shortness we call Caesar, changed its position in 
space in relation to a second group of sensible impressions, 
which we call the Rubicon, in such wise as to be perceived 
by one and the same spectator first to the right of the latter 
group and then to its left. I answer with no less obstinacy: 
that this group was the same group on the left as on the 
right, that is to say that it has changed its position, — this 
does not lie in the simple data of observation, but is a 



Chap. V.] MATHEMATICS AND IDENTITY. 



297 



hypothesis which covertly introduces under a connected 
and continuous alteration of the appearance a permanent 
substratum with merely changing relations. Whenever in 
recounting an event we speak of any sort of movement in 
space, we are giving not our perception, but a hypothesis 
about it. That one and the same real a 1 passed through 
one after the other the places m, n, fi, is not a fact we have 
seen ; the fact perceived is only that in successive points of 
time similar appearances a were observable in successive 
points of space. One who was under no necessity to ex- 
plain this fact to himself by the hypothesis of a permanent 
subject, could not venture to affirm the proposition i a has 
moved/ as a description of the facts, but merely as a con- 
venient mode of expression, having in relation to fact no 
significance whatever. If he denies himself this intro- 
duction of certain points of view into the interpretation of 
the content of perception, then he must acknowledge all 
synthetic judgments a posteriori, all judgments indeed of 
whatever kind, to be inadmissible, and instead of a re- 
counting of past facts there remains in truth merely the 
possibility of recalling in memory a series of perceptions, a 
reproduction of the raw material, out of which judgments 
might be formed, if only such a proceeding were allowable. 

352. Turning to the question of the discovery of mathe- 
matical truth, we shall not dispute the validity nor yet the 
importance of the principle of Identity, but we must dispute 
its fruitfulness ; we must insist that if it were the only prin- 
ciple we had to start from, mathematical truth could never 
be discovered at all. It is no doubt true that in any pro- 
position affirming equality or inequality, a — b or a > b, we 
have always to assume the validity of the principle of iden- 
tity, according to which a = a, and b = b, in other words 
that every quantity which we desire to bring into any rela- 
tion with other quantities, is identical with itself, for 
obviously every such comparison of different quantities 
1 [< Dasselbe reale a.'] 



298 



THE A PRIORI TRUTHS. [Book III. 



loses its meaning if the quantities compared may have an 
unlimited variety of meanings. Here the principle of iden- 
tity has a validity which is manifest enough and is the 
necessary security for truths of whatever kind. But it is 
precisely from this point of view that least attention has in 
fact been paid to it ; that which has been more especially 
emphasised is that very different application of the principle, 
by which the two quantities compared are pronounced equal 
to each other. It is in this application of the maxim of 
identity that philosophers have found not only the guarantee 
of truth, but when repeated in frequent succession through 
a long chain of such equations, a fruitful method for its 
discovery. 

I cannot think that either the one of these contentions or the 
other precisely expresses that which is intended. Equations 
either as in ^4=2 express simply the definite quantitative 
value which is arrived at by an act of calculation as applied 
to a given quantity, or else they express the fact as in v ab 
= Va. that certain operations, different in form, ap- 
plied in a prescribed order of succession or of connexion to 
any given quantities within defined limits will give identical 
results. Now in both these cases the value of the entire 
mathematical process depends not solely upon the dis- 
covered equality of the result but rather upon the fact that 
different paths have led to the same goal, that is to say that 
it has been found possible to affirm the equality of different 
things. If I am answered that the quantitative values of 
the two different terms in the comparison have not been 
made identical as an after result of that operation but were 
so always, and that the identity was merely concealed under 
the different forms in which the two were originally pre- 
sented, or that the one form of expression merely sets the 
problem of which the other gives the solution, — such a 
reply expresses precisely my own view, only that it takes as 
self-evident that which I cannot regard as being such. 

For whence do we derive our confidence in the possi- 



Chap. V.] NO INFERENCE B Y BARE IDENTITY. 299 

bility of one and the same self-identical value being pre- 
sented under different forms ? Certainly not from the law 
of identity alone ; for it contains not the slightest hint of an 
antithesis between Form and Content or Form and Value ; 
nor supposing that we derived our idea of such an antithesis 
from some other source, could the law of identity even then 
tell us anything whatever about it. It could only tell us 
over again, every Form is identical with itself and every 
Value with itself. That one and the same Value can be 
present under different forms, it could never affirm, because 
it could never fix any limit to the validity of such a propo- 
sition except one which would reduce it to a barren 
tautology. For to the question what different forms of 
expression designate identical values, it could only answer, 
those in which one and the same identical value is con- 
tained. 

I need not here enlarge on the fact, that it is in this 
possibility of affirming the equality of the different, and not 
in the bare application of the logical law of identity as such, 
that the motive force of all fruitful reasoning in mathe- 
matical science is to be found. We should never get any 
further, if we could never subsume under the subject of a 
given major premiss anything but a term absolutely identi- 
cal with it ; we do make progress just because by means of 
innumerable substitutions, by a process of analysis on the 
one side and recombination on the other we are able to 
bring a quantity given us in the form a into the form b, and 
thus to subsume it on any occasion under such a major 
term, as then enables us further by known methods of 
calculation to give it a predicate which was not deducible 
from it in its original form. Everything turns therefore on 
our right to affirm identity of the different, and this right 
does not follow, at all events as an immediate consequence, 
from the purport of the law of identity. 

353. The remainder of my argument here must be taken 
in connexion with the considerations which I urged when 



300 THE A PRIORI TRUTHS. [Book III. 

dealing with the subject of pure Logic as to the nature of 
judgments synthetic in form but identical in content. I 
there made allusions to Kant, who in endeavouring to 
prove the presence of synthetic judgments a priori in all 
branches of reasoning included under that category the 
arithmetical judgment 7 + 5=12. My object at that point 
was to insist on the identity of content which must neces- 
sarily obtain in any true proposition between the subject 
and the predicate taken in their entirety. I was dissatisfied 
that this point should not have been more expressly insisted 
on by Kant, but I then reserved the right to revert again to 
the truth which his doctrine contains (§ 58). Kant held 
that we could not possibly recognise in the predicate 1 2 the 
solution of the problem expressed in the subject 7 + 5, 
without an act of Perception 1 . Perception alone, that is, 
he insisted, can establish for us that the identity required 
between the two sides in order to the correctness of the 
equation is actually the fact. Considered for purposes of 
illustration, indeed, I think that Kant's example was not 
happily chosen, because it does not bring the formal differ- 
ence which exists between the subject and the predicate, 
and upon which stress ought to be laid, into sufficiently 
clear prominence. It is true, indeed, that 12 is not merely 
another name for 7 + 5, but expresses something quite dis- 
tinct, viz. that the same quantity which is produced by the 
addition of 7 and 5 also occupies a place as a definite term 
in the numerical series between n and 13. But then the 
simplest idea which we can form of that series itself is to 
conceive it as arising out of repeated additions of the unit, 
that is to say out of the very same operation through which 
7 and 5 themselves were put together. So that we conceive 
the left side and the right side of the equation equally as a 
sum of units and we merely analyse, on the left side, into 
two steps, as the idea of a sum allows us to do, that which 
on the right we take as a whole. 

1 ['Anschauung.'] 



Chap. V.] 



THE FACT OF QUANTITY. 



301 



On the other hand such a formula as 7 + 5=4 2 — 2 2 , 
though not in fact expressing any more completely than the 
other that which is essential in Kant's thought, yet would 
have better illustrated the point that there are various ways 
by which we may arrive at one and the same quantitative 
value. For that which all turns upon is in fact nothing 
more than the assertion which is contained in the sign of 
addition, — viz. that quantities can be summed so as to 
compose another and a homogeneous quantity; a propo- 
sition the importance of which we may once more be 
tempted to ignore, because it seems to us self-evident and 
a mere identical proposition defining the nature of nume- 
rical quantity as such. And so it undoubtedly is, but how 
do we arrive at this piece of self-evident knowledge ? Not 
every ideal content will submit to the same operations ; we 
cannot add red and green together and produce blue ; the 
notes c and d do not admit of being summed so as to pro- 
duce a third note x, such as to stand higher than d in the 
musical scale by the interval c, just as 1 2 stands higher than 
5 in the series of numbers by 7. 

But here the question may be asked in surprise, what 
does this last remark lead to ? Of course, it may be said, 
mathematical operations can only be applied to quantities, 
whose nature it is to admit of them, and not, or at all events 
not immediately, to impressions which are qualitatively 
different. But this is really to be blind to what lies under 
our very eyes. This very fact, that there is such a thing as 
quantity to be found in the world of ideas, while yet our 
thought itself is not bound, on pain of not taking place at 
all, to be the thought of just these comparable quantities — 
this very fact is a fact of immediate perception, which if it 
were lacking to us, could be as little supplied through 
logical operations working on a different set of ideas, as 
could the conception of qualitative resemblance if the world 
of ideas presented no comparable impressions of sense such 
as colours or sounds. The proposition therefore that 



302 



THE A PRIORI TRUTHS. 



[Book III. 



quantities can be summed is undoubtedly an identical 
proposition ; but that the subject and predicate of that 
proposition appear as valid in the world of ideas, and that 
it has quite a different value from the equally identical pro- 
position, all wooden iron is wooden iron, — this does not 
follow from the principle of Identity. It is not then the 
bare logical principle of Identity, but the perception of 
quantity, the peculiar nature of which makes it possible to 
frame a countless number of propositions in content iden- 
tical yet in form synthetic, which at once guarantees the 
truth of arithmetical reasoning and is the source of its 
fruitfulness. 

That which might here be added in the interest of 
mathematics, I must pass over; with regard to the logical 
question I confess myself in entire agreement with Kant in 
a further point, namely in maintaining the pure or a priori 
perception of numerical quantity in the sense of the word 
a priori explained above. It is true that neither the idea 
of quantity as such, nor the more defined conception of its 
capability of being summed, nor finally any one arithmetical 
proposition, ever enters into our consciousness without 
being occasioned, and the occasion can always be traced in 
the last resort to an external stimulus. We think them 
only when we are led in one way or another to frame the 
idea of numerable objects. But, when the occasion arises, 
we do not learn that 7 + 5 = 12 from the content of this 
perception, in such a way that our knowledge of the truth 
in question would gain in certainty with every fresh con- 
firmation by subsequent experience \ but as a matter of fact 
the single presentation of the idea 7 + 5, no matter whether 
mediated through external perception or not, is sufficient to 
teach us its identity with the term 1 2 once for all and as a 
fact of universal validity. And supposing that we found 
when we came actually to count external objects in a variety 
of further instances that our arithmetical proposition was in 
some cases confirmed and in others not, we should certainly 



Chap. V.] SYNTHETIC JUDGMENTS A PRIORI. 303 



all of us, even the most decided adherents of empirical 
theories, agree to correct not our arithmetical proposition 
by our countings but them by it. 

354. The case is perhaps still clearer if we turn to 
geometry. As to Kant's particular instance of a synthetic 
geometrical proposition, a straight line is the shortest line 
between two points, I have alleged similar scruples as in the 
case of his arithmetical example just discussed. Here again 
the example is not happily chosen, because we have no 
other direct standard of measurement for the conception of 
distance which is contained in the predicate ' shortness ' but 
the straight line itself, and thus the proposition suggests 
before everything else the complete identity of its subject 
with its predicate. And such identity does undoubtedly, in 
respect of their content, exist; the proposition would not 
otherwise be true at all ; but once more, how do we establish 
that identity? By connecting the two points through a 
something which we say is ' between ' them. Now it is clear 
that this expression does not mean merely that the two 
points are logically designated as not identical or as merely 
in some way or other different, for that is equally the case 
with green and acid, out of which a proposition of this kind 
could never be formed. Nor again is it merely that they 
can be compared, for so — again with no such consequence 
following — can green and red. What it means is that they 
are connected in a manner completely sui generis, which is 
thinkable and has a meaning to us through an original 
faculty of spatial perception and so only, and which in the 
absence of such perception could never be made intelligible 
to us through any logical operations working on a content 
of a different kind, and of which, finally, even now when it 
is familiar to us all, no form of words, unless tacitly implying 
such spatial perception, can by any possibility give us a 
clear idea. 

Other instances of Kant's are more expressive. Take, he 
says, the proposition that two straight lines cannot enclose 



304 



THE A PRIORI TRUTHS. [Book III. 



a space, or, therefore, make up a figure, and try to derive it 
from the conception of straight lines and the number two ; 
or again the proposition that out of three straight lines a 
figure can be formed, and try to deduce this in the same 
way from the conceptions it contains. Your labour is all in 
vain, you find yourself driven to have recourse to Perception, 
as Geometry in fact always does. These words remain true, 
even though a slight inexactness in the form of expression 
may offer a handle to controversy. The complete subject, 
in the second example, to which the predicate of c forming 
a triangle ' belongs, is not simply ' three straight lines ' \ the 
lines must be in the same plane, they must not be parallel, 
they must admit of being produced at pleasure. Again in 
the first of the two examples, we have no right to require 
the impossibility of the closed figure to be deduced from 
the isolated conceptions of the number two on the one hand 
and of the straight line on the other ; we must begin by 
representing ' two ' as the number of the lines, and the lines 
themselves as included in the same world of space. If we 
add these fresh points, the predicate will be seen in both 
cases, though not in both with equal obviousness, to follow 
identically from the subject when taken in its full meaning, 
and so the identity of their contents, which is essential to 
the truth of the proposition, will be established. 

But this mere matter of statement does not alter the 
question at issue. All these conditions, that the lines must 
belong to the same plane, must not be parallel, must be 
capable of being produced, have absolutely no meaning 
whatever, unless we assume the spatial perception to begin 
with. It is this and nothing else which is our evidence that 
anything answering to those expressions is to be met with in 
the world of ideas, and which alone assigning a thinkable 
meaning to the complete subjects of the propositions in 
question, gives a reason in so doing for the predicates 
identical with their subject, which in each case they contain. 
These propositions then are undoubtedly identical propo- 



Chap. V.] PRINCIPLES HOW EXPERIENCED. 305 



sitions, although under a synthetic form ; but that their full 
content and the manifold relations contained in it exist, is 
not due to the principle of Identity. That is to say, it is not 
by means of the principle of Identity that we can pass from 
one form of expressing a geometrical fact to another equiva- 
lent to it ; rather it is the peculiar nature of space which 
makes it possible for identical facts to be variously expressed. 
It is upon this fact, and more especially upon the unlimited 
power we possess of bringing any given figure in space by 
the help of artificial combinations under fresh mathematical 
points of view or fresh general ideas, and thus constructing 
predicates for it, which were not contained in our original 
conceptions of the figure, — it is upon this fact, and not 
upon the mere application of the bare principle of Identity 
as such, that the fruitfulnes.s of geometrical procedure 
depends. 

355. At this point I may expect the criticism that my 
argument has brought me to a different conclusion from 
that which it was aiming at. I began I shall be told by 
maintaining the necessity, in order to any extension of 
knowledge, or to the discovery of the laws of nature, of 
synthetic principles a priori. And now I am invoking the 
aid of Perceptions to supply both subject, predicate, and 
copula of the judgments in which we express those 
principles, a proceeding which seems after all to amount 
to no more than the not very helpful proposition, that we 
cannot think without having some idea of that which is to 
be the content of our thought ; the fact still remains that 
the object in question is given not by thought but to thought, 
in a manner not essentially different from that in which 
every other object of consciousness is given, namely through 
experience. 

In regard to this last point I repeat once more in one 
word what I have said already, that all knowledge whatever, 
whether innate or not, which as a matter of fact whether 
constantly or upon occasion makes its appearance in the 

Logic, Vol. II. X 



3o6 



THE A PRIORI TRUTHS. 



[Book III* 



consciousness of any one, is for him in the broader sense of 
the word, an object of experience. And further we have 
admitted from the first that no one of the principles which 
we regard as innate, can be operative in us even in the 
sense of a major premiss unconsciously followed in our 
judgments, until an incitation so to follow it comes to us 
in experience, while it can only become in the full sense an 
object for our consciousness through a definite act of reflexion 
upon those applications of it which have already been made 
unconsciously. In this sense then I have no objection to 
offer if any one insists on calling the apprehension of 
a priori principles itself an inner experience ; I only regard 
it as a perfectly barren contention. Nor can the con- 
troversy between the a priori and the empirical view turn 
on the further point that the latter ascribes to outward that 
which we would rather attribute to inner experience. For 
this antithesis simply does not exist ; whatever notions we 
may form about a supposed external world, our experiences 
can only be of the representations of it in our own minds, 
of the order and connexion of our own ideas. Upon this 
point I may be allowed to be brief. In Germany at all 
events the fallacy which has been imported from abroad 
does not yet find favour, that by measuring the solid and 
superficial angles of material forms we can confirm the 
propositions of geometry, or discover any others than those 
which we can develope with our eyes shut from relations 
assumed to exist between mere points of space. We are 
still aware that such measurements, supposing that we make 
them, relate directly not to the nature of the bodies which 
fill the space in question but to properties of the space 
which they fill ; finally that they can only be made at all by 
the aid of contrivances and methods which are all founded 
to begin with upon the essential order and regularity of our 
spatial perception, and that we can never therefore employ 
the process of measurement to test this our geometrical 
knowledge by the standard of a knowledge which has a 



Chap. V.] 



NATURE OF SELF-EVIDENCE. 



307 



different and independent source, but that so far as we do 
employ it we are merely bringing a particular case of spatial 
Perception under the laws of geometrical Perception in 
general. 

Thus the difference between us and our opponents comes 
back merely to this, that to us the simple principles of 
geometry, that every straight line may be produced to 
infinity, that the opposite angles of two straight lines inter- 
secting one another are equal, that the sums of any two 
adjacent angles are equal, — such principles are to us truths 
which once presented to thought are valid always ; whereas 
in the view of empirical philosophy each particular appre- 
hension of them must in consistency be regarded as a 
psychical fact and nothing more, as to which there is no 
certainty whether it will recur in a similar case or not, and 
of which therefore universal validity can never be established 
as true, and can only be established as probable on the 
strength of the agreement of a very large number of instances 
in which as a matter of fact it has so recurred. 

356. I must once more summarise my position in relation 
to this point of view. In the first place the contention that 
every truth of whatever kind requires this test of experience 
in order to be received as universally valid, would contradict 
itself. For on the one hand it must itself come under its 
own rule and by consequence cease to be universal ; on the 
other hand, as we have already seen, without the assumption 
of the unconditional validity of some absolutely certain 
principles not drawn from experience the very deliverances 
of experience itself could be no one more probable than 
another \ 

On the possibility of an immediate knowledge therefore 
of some universal truth all certain belief depends, that of our 
opponents no less than our own ; the difference between us 
can only be as to what the truths are which we hold to be 
accessible to this form of knowledge. But it is self-evident 

1 [§ 330.] 
x 2 



3 o8 



THE A PRIORI TRUTHS. [Book III. 



that in the case of truths which are to be recognised 
immediately as universally valid, their sole credentials must 
be the clearness and strength with which they force them- 
selves upon consciousness and at once claim recognition 
without constraining it by any process of proof. Now 
any one has perfect liberty to allow this claim or to 
resist it ; it is open to every one whether in all honesty to 
distrust the self-evidence with which this or that object of 
knowledge presents itself to his consciousness, or to insist 
(at all events for sophistical purposes) that no self-evidence 
in the world affords a proof of the truth of the thing 
evidenced ; only in the latter case he must allow that a like 
vein of sophistry may contest the validity of any process 
of proof whatever and of his own contention along with the 
rest. 

This sort of idle disputation for disputation's sake we 
may leave to itself; the former more honest variety of 
scepticism on the other hand is not without its justification, 
for undoubtedly that state of repose and peaceful equilibrium 
of the mind, in which the self-evidence of knowledge, 
regarded as a psychical fact, consists in the last resort, may 
also be produced by conjunctions of ideas of by no means 
universal validity. These false forms of self-evidence we 
have admitted to exist, and the logical processes have been 
given through which we seek to free ourselves from such 
illusions. These processes all resolve themselves into this — 
by shaping our investigations in various ways, adopting 
various starting-points and various methods, we arrive at 
separating from a subject *S to which it is our object to 
ascribe a predicate P } all associated ideas x, not really 
contained in £ but secretly affecting our conception of it, 
which might create in us the impression that our P which 
in fact belongs only to S + x is an invariable attribute of £ 
as such. Our method does not always assume the form of 
a direct proof; the proposition that a straight line may be 
produced to infinity is too simple to admit of any argument 



Chap. V.] 



INTUITION, 



309 



except one which brings us back by a complete tautology to 
immediate Perception ; in other cases again proof will take 
the apagogic form of a reductio ad absurdum, a form of 
argument which does not deduce the truth of the given 
proposition from some other acknowledged principle but 
merely establishes the impossibility of denying it. When 
this has once been accomplished we regard the proposition 
in question as a truth of universal validity, needing no 
empirical confirmation from particular instances in which it 
is found to hold, but on the contrary standing over against 
all particular instances as certain a priori. We do not deny 
the possibility that this trust in reason may now and again 
deceive us ; but we should not surrender the presumption 
in favour of a principle thus arrived at being true merely 
because it is possible to distrust it ; we shall hold fast to it 
until either the results to which it leads involve us in con- 
tradictions, or until some other truth becomes plain to us, 
from which we are able to understand how a proposition 
now seen to be false came to present the appearance of a 
self-evident truth. 

357. There are various points here which still need 
elucidation. The terms in w T hich in the Kantian school 
pure Intuition 1 has been spoken of in contradiction to 
Thought, have led to its becoming associated with the idea 
of a peculiar and somewhat mysterious form of procedure 
through which the apprehending mind accomplishes some- 
thing which is impossible to its discursive thought. The 
obscurity which attaches in consequence to this idea is due 
to this, that in fact it is just in the case of Intuition that no 
sort of procedure consisting of the connecting of various 
single acts is describable, whereas there is one in the case 
of Thought. The attitude of Intuition towards its content 
is that of passive receptivity, and its work is done so com- 

1 [' Anschauung.' ' Perception ' is usually a better rendering of this 
word than 'Intuition'; but the latter is preferred in this passage for 
obvious reasons.] 



3io 



THE A PRIORI TRUTHS. 



[Book III. 



pletely at a single stroke, that no steps or stages in it can be 
distinguished or could be described. This must not be 
misunderstood. 

When geometrical intuition teaches us that two straight 
lines intersecting each other can only have one point 
common to both, there does undoubtedly take place, re- 
garding the act as a psychical event, a certain succession of 
ideas, which we might describe if in any particular case it 
were exactly known to us. We might explain how we first 
think each of the two straight lines in itself, then place 
them in the same plane, make them from a parallel position 
converge, follow each to the point of section and then 
beyond it, — all this we can describe, but this is not the 
geometrical intuition itself : so far we have only brought 
into consciousness all the different points which go to make 
up the relation in question, and now intuition pronounces 
on these points of relation, as by a single instantaneous 
revelation, — the two straight lines can only have one point 
in common. How this final step is accomplished, the 
immediate apprehension of the necessary truth which is 
implied when once all the members of the relation are com- 
pletely given, is a point upon which certainly at present, 
and in my judgment no less certainly for ever, any further 
psychological analysis is impossible. It is only in this sense 
of absolutely immediate apprehension that I have here 
employed the term intuition, and it leads me to a further 
observation as to the meaning of the expression a priori as 
applied by us to intuition. 

I have explained before why it is that knowledge must 
necessarily consist not in the mere passive reception of 
impressions but in a reaction, the form of which reaction 
will depend on the nature of the mind which is stirred to 
it. I did not conceal my agreement with Kant in account- 
ing Spatial Intuition as a form of such reaction, and there- 
fore as a priori or innate in the sense in which that term 
may legitimately be used. For the question before us how- 



Chap. V.] 



'INNATE' IDEAS. 



ever this point is of no importance. It is not because the 
idea of space is innate in us, that we are m a position to 
frame universal propositions in geometry, which once 
thought are valid always ; if it were at all intelligible with- 
out any such hypothesis how the idea of a particular com- 
bination of spatial points of relation could arise in us purely 
through external impressions, still, in presence of such an 
idea, the immediate apprehension of the universal truth 
contained in those relations, which is the service of intui- 
tion, would be not more inexplicable (though it would be 
equally inexplicable) and not less possible than if those 
same points of relation could only be brought into our 
consciousness by the help of an innate mode of reaction 
and spontaneity in the mind itself. I therefore reserve the 
question of the a priori, in the sense of the innate character 
of spatial Intuition, with any further question which may 
arise out of it, for the Metaphysic, and apply the term a 
priori to spatial intuitions in a restricted sense only, viz. to 
indicate that they are not derived by a process of induction 
or summation from particular instances which exhibit them, 
but are thought to begin with as truths of universal validity, 
and are thus prior to the particular instances in the sense of 
being rules by which they are determined. 

358. This brings us to the last point which we have here 
to consider. Philosophers have spoken of pure Intuitions 
as an innate possession of the mind, in terms which could 
not but lead as a natural deduction to the idea that all 
truth which rests upon any such intuition is also an in- 
tellectual treasure always at hand, which we take with us to 
experience, and through which we judge it. And in fact 
Locke made use of this deduction as an argument against 
the doctrine of innate Ideas. It needs however only a 
brief consideration to see that such a deduction is illegiti- 
mate. Every one who speaks of innate knowledge includes 
in it most certainly mathematical truth, but mathematical 
truths had all to be discovered before they were known, 



312 



THE A PRIORI TRUTHS. 



[Book III. 



and the universally innate possession of a spatial intuition 
was not the same thing as the possession of a knowledge 
of geometry. But the most elementary of these truths were 
discovered as soon as ever the mind was drawn to turn its 
attention away from the infinite variety of figures presented 
by bodies in space which surround us in the world of per- 
ception, to the simplest relations which are contained in all 
of them alike. Then at once the truth of each several 
principle one by one sprang to light self-evident and self- 
proved, just as Plato so admirably represents it in the 
Meno, only that it was superfluous to refer us to a previous 
state of existence from the memory of which this sudden 
emergence of knowledge was supposed to come, inasmuch 
as there also the conviction of the certainty and necessity of 
the truths which there were given to intuition in a universal 
form could only have arisen in the mind through the same 
immediate act of apprehension by which in our life here we 
recognise it in particular instances. 

It is still easier to understand how it is that the more 
complicated mathematical relations should have had to 
wait for their discovery, and that an immense tract of 
ground should always remain before us, in which new 
discoveries are to be made. The consequences which 
follow from simple mathematical principles become science 
only by being deduced from them by reflexion, and this 
operation involves a labour of a most extended and con- 
stantly progressive character, the application of processes of 
exact definition, of analysis into varied elements, of syn- 
thesis into well-defined forms, to abstractions made by the 
mind itself, and this in order to establish even the subjects 
of the propositions required, the predicates being obtained, 
it may be, by processes no less elaborate. 

Paradoxical therefore as it may seem we must disabuse 
ourselves of the false idea that the world of the self-evident 
lies of itself plain in its self-evidence before us, and that all 
we have to do is by the help of this comfortable possession 



Chap. V.] MECHANICS 'PRESCRIBES' LAWS i 313 



of a self-evident truth to go on to subdue the intractable 
world of our perceptions. The fact is that even universal 
truths, for the apprehension of which the mind requires 
nothing outside itself, have yet to be found by searching, 
have to be abstracted and separated off from among the 
measureless host of ideas which form the world of conscious- 
ness. Nor can we even expect that the very simplest of 
innate truths, the highest principles of all, will be revealed 
to consciousness first of all by this process of self-reflexion. 
On the contrary, their first appearance is invariably occa- 
sioned by some particular instance which exemplifies them, 
or some particular case presented by perception or by 
imagination that the mind may pass judgment on it. But 
it may happen that our perceptions may be of such kind as 
never to present to us the case required in its purity, and 
in the same way to debar the imagination from conceiving 
the idea of it, and this though if once it were presented to 
consciousness, the mind would at once feel the conviction 
awakened of a truth of the most universal and fundamental 
kind, and would judge accordingly. Thus then it may be 
an extremely difficult task for knowledge, to remove all the 
obstacles which the actual connexion of our ideas, imposed 
upon us by experience, plants in our path, and to fight its 
way through to the knowledge of the self-evident. 

359. In mathematics, where the matter of investigation 
can most easily be separated from the real objects to which 
it is attached in experience, it has speaking generally been 
possible to advance from the simplest truths to their deri- 
vative consequences, although, in spite of this, the fresh 
knowledge has afforded new and more comprehensive 
expression even for the principles which were known before. 
It has been otherwise in the science of mechanics, which 
applying itself directly to actual occurrences, seeks to 
prescribe laws to the interactions which obtain between real 
things. I use this much criticised expression 1 of Kant, in 
1 [' Vorzeichnen ' (to prescribe).] 



314 THE A PRIORI TRUTHS. [Book III. 



order to reduce the objections which have been made to it 
within their proper limits. No one could have intended it 
to mean that human reason can invent laws at its own good 
pleasure which nature is bound to follow. But supposing 
the idea of a relation between different elements to be 
presented to us in so simple a way as to exhibit an instance 
of the perfect purity required, in which true laws of nature 
are seen producing their simplest result, with no multiplicity 
of extraneous conditions to obscure it, why should it not be 
possible in such a case for reason, itself a member in the 
system of the world in which these operations take place, 
to have an immediate apprehension of the result in which 
the relation supposed must necessarily issue ? This is not 
to thrust its own subjective laws upon nature, but to detect 
the real laws of nature herself, which become to it binding 
rules which it brings with it to the confused tangle of 
separate events wherewith to estimate and interpret them. 

In this sense pure mechanics is an a priori science ; it is 
quite true that many of its principles may have been first 
suggested and the enquiry after them occasioned by experi- 
ence, but it was not by the testimony of repeated perceptions 
that they were discovered and reduced to the exact form of 
a law, it was by an operation of thought, apprehending with 
the clearness of immediate vision the self-evident law in an 
instance where it is presented in its purity, and in compli- 
cated cases finding means to reduce them to a similar 
simplicity. This is commonly expressed by saying that 
within its own province mechanics is an absolutely demon- 
strative science, which from pre-suppositions of its own 
creation evolves necessary conclusions with irresistible logic ; 
but that, to compensate for this it has in relation to experi- 
ence only a hypothetical validity, that is to say it is valid 
only on the assumption that real things exist which admit 
of being subsumed with perfect exactness under the 
conceptions from which its conclusions are drawn. 

But such language allows too much to an unjustifiable 



Chap. V.] THE ABSTRACTIONS OF MECHANICS. 315 

scepticism as to the tenableness of the hypothesis, and does 
not really answer to the facts. For the science of mechanics 
did not spring up in some meditative consciousness, playing 
with possibilities before experience existed, it arose under 
the persistent pressure of experience which called for expla- 
nation. The abstract universal conditions, from which in 
mechanics we derive definite consequences, are not Proble- 
matic schemes of something which might perhaps be found 
in reality, but reductions of that which is Assertorically 
given in experience to a form in which its validity is 
universal. 

But that reduction to a universal form was necessary by 
reason of the one actual postulate with which the science of 
mechanics stands and falls, that a uniformity of law does 
hold good in the world of events. If this assumption is 
justified, and if there are many elements ABC... oper- 
ating together in the order of nature, each under various 
forms a a> a 2 . . . ß ß 1 ß 2 . . ., finally all of them in varying 
relations M N . . . each of which again may assume different 
values /x fx 1 . . v v 1 . ., — then any single event must be the 
joint effect of many single laws, each law concerning two 
elements A B only and their relation M, and determining 
the particular operation E which results from these data, 
and which in turn will change to e e l . . ., as A B and M 
pass through their several changes in form or in value. 

It may be that experience never affords a perfectly pure 
instance of one such single law ; still it would be folly to 
find fault with mechanics for speaking of motion in the first 
instance without taking account of resistance which never- 
theless invariably attends it; or of a homogeneous mass 
which is nowhere to be found, or finally of a perfectly rigid 
body, whereas perception presents us only with bodies 
which are elastic, yielding, of various degrees of hardness. 
It will be time enough to take account of the influence of 
these secondary conditions when we have learnt the uni- 
versal laws upon whose consequences they exert their 



3i6 



THE A PRIORI TRUTHS. 



[Book III. 



modifying operation ; but even supposing that the theory of 
the resisting mediums, of the specific qualities of matter 
and its molecular properties, were never to reach the simple 
clearness of the other departments of mechanical science, it 
is certain that a philosophy of nature which was not even 
acquainted with the laws of the simple and pure cases from 
which every individual mixed case varies by a determinate 
amount, would be attended by still less success. For it is 
by no means for the mere convenience of shortening our 
procedure at the cost of its exactness, that we ignore the 
special peculiarities of the concrete instance, and begin by 
looking for the law of a universal and abstract instance ; 
our assumption of the presence of law in actual events 
involves as a necessary and objective consequence that the 
joint operation of many elements is made up of the several 
operations, which each pair of elements, combining in a 
specific relation, generate on their own account, and which 
they alter in accordance with a fixed law as this relation alters. 

360. The empirical content and course of our perceptions 
has rendered it by no means an easy task for mechanical 
enquiry even to form the ideas of the simple and pure cases 
upon which an immediate intuition of the truth could at 
once pronounce a judgment self-evident and universal ; on 
the contrary, it is here more than anywhere else that experi- 
ence has exerted the injurious influence already alluded to, 
drawing the mind away from the apprehension of the 
universal and the unconditioned, by constantly introducing 
to it the particular and that whose validity is conditional. 

The entire period of antiquity passed away without the 
conception of motion, the central point in mechanics, 
having been reduced to a form simple enough to be 
immediately apprehended by the mind in its abstract cha- 
racter. Three great examples of motion were presented by 
experience to the imagination, the perpetual motion of the 
heavenly bodies, the rapidly ceasing motion of terrestrial 
bodies caused by external impulse, finally the energy of 



Chap. V.] HISTORY OF 'MOTION' AND 'MASS.' 317 

living beings, originating within but after a while wearying. 
The mind of antiquity never succeeded in separating the 
simple process in which all motion consists, continuous 
change of place, from the conflicting peculiarities of these 
different classes of instances in which it occurs. The 
phenomenon was never disentangled from certain assumed 
causes of it ; the course of the stars was represented as a 
divine motion exalted above the general laws of nature, or 
else the motions of terrestrial bodies were attributed to an 
extraneous necessity and thus degraded to a position below 
the due and natural order of things. Add to this that the 
analogy of the wearying of human activity led men to regard 
cessation of all motion as such as the natural and self-evident 
law, its eternal continuance as a divine exception. 

It was reserved for a much later epoch to conceive the 
essential features of all motion of whatever kind, as consist- 
ing simply in a relation between the three elements of 
velocity, duration of time, and space traversed, and by the 
modest formula s = vt to lay the foundation of a scientific 
theory of motion. That formula once given, the law of the 
persistence of motion followed of itself : for although the 
discovery of the law was due to a generalisation from 
particular results obtained by experiment, showing that 
motion always lasted for a longer time in proportion as all 
external hindrances were removed, still no one doubts 
that directly it was discovered it expressed a tardily appre- 
hended necessity of thought. That there is such a thing 
as motion had to be learnt from experience, but if it exists 
or is to exist at all, the idea of its persistence becomes a 
necessary postulate in order to make it even a possible 
object of intuition 1 . 

Similar difficulties had to be encountered in forming the 
conception of mass. The bodies with which we most 
ordinarily deal, whether solid or fluid, were observed to 
follow the downward tendency of weight, whereas vapours 
1 [See § 247.] 



3i8 



THE A PRIORI TRUTHS, 



[Book III. 



and fire tended upwards; thus the idea arose of two 
opposite impulses, both belonging essentially to the nature 
of bodies, but leading away from one another in two 
opposite directions, directions which might indeed have 
been correctly distinguished by a qualitative distinction 
of the ultimate points towards which they tend, but which 
were in fact confounded with the unintelligible antithesis 
of an above and below in an absolute space. It took a 
long time before the combination of more extended ob- 
servations was able to compensate for the one-sided 
character of the facts as experience at first presents them, 
and to show that neither the direction nor the intensity 
of weight-pressure was everywhere uniform. Not till then 
did the natural idea make its way that the beginning of 
any new motion whatever must necessarily require some- 
thing to determine its direction a fronte or a tergo, in 
the way of attraction or repulsion along a straight line, 
that is to say, that it takes its origin always from an 
interaction of different elements in space, and that the 
amount of such interaction depends on the quantities of 
a homogeneous real existence which are united in each 
one of the elements in question. The idea of mass, again, 
which was thus arrived at, in which regard is had only 
to the amount of the resistance of inertia on the one 
hand which real existence in space offers to any motion 
which is demanded from it, and to the magnitude of 
the power with which it enforces every motion of other 
elements which originates from itself, — this mechanical 
conception may very well stir new questions to which 
philosophy would have to find an answer; still when 
once a regular order of natural events is given or is 
assumed, in which each single event is taken as the 
condition determining the definite degree of another event 
following upon it, it is easy to see that such a conception 
involves as a self-evident postulate the commensurability 
of all real elements in regard to the magnitude of the 



Chap. V.] SYNTHETIC TRUTHS IN MECHANICS. 319 

effects they may be expected to produce, a principle which 
is expressed in the conception of Mass. But how great 
the power is which one-sided and partial observation 
exerts over our conceptions, is attested by the difficulty 
which the common imagination finds even in the present 
day in believing in the possibility of the Antipodes, and 
again by the errors of certain schools of natural philosophy, 
to which not indeed the eternal downward motion of 
the philosophy of antiquity, but still the concentric pressure 
of gravity, formed so essential a part of the general notion 
of material substance that the idea of mass without weight 
always seemed to them a contradiction. 

Here I must break off ; but any one undertaking to 
write the history of the development of mechanical con- 
ceptions would find it a suggestive task, instead of being 
content perpetually to repeat how we have come simply 
through the connexion of particular experiences to our 
knowledge of natural law, to go on to trace and explain 
how at first the partial and one-sided character of those 
experiences forced upon men's minds a number of false 
ideas, and hindered them from arriving at an earlier ap- 
prehension of self-evident truths. 

361. There are conflicting opinions as to the logical 
character of the simplest mechanical principles. Just 
because they concern in the first instance not actual bodies 
but a certain postulated subject-matter whose nature is 
wholly determined by our definitions of it, we either 
consider ourselves bound to look upon them as analytical 
judgments the truth of which is guaranteed by the law 
of Identity, or else we regard them, even as taken in 
their purest and most abstract form, as still synthetic 
and therefore mere probable hypotheses, whose truth can 
only be established by their agreement with experience 
and the complete internal harmony of the conclusions 
to which they lead. 

My own judgment in this controversy can be no other 



320 



THE A PRIORI TRUTHS, 



[Book III. 



than that already given in reference to the kindred prob- 
lems in Arithmetic and Geometry, but I must content 
myself with briefly indicating my point of view without 
developing it in all the detail which might be desirable. 
In general I might express my position thus : the two 
given data A and B, as to whose connexion a mechanical 
judgment is to be affirmed, are not given to us merely 
one by one ; our ideas of them are only intelligible and 
are only understood in and through a single Intuition, 
which embraces both together and which determines also 
in one and the same act the relation between them. 

Let us, to begin with, turn once more to an example 
from arithmetic. The proposition 30-30 = 0, we shall 
be disposed to refer immediately to the principle of 
Identity ; nevertheless all that that principle tells us, taken 
simply by itself, is that 30 = 30, — z a — —3 a > an d finally 
Z a ""Z a — Z a ~Z a \ that this last expression = o, we can 
only maintain on the strength of a direct intuition of the fact 
that there are two operations lying within our power, the 
addition of a to a and then the subtraction of a from 2 a, 
which exactly cancel each other, and in the repetition of 
which an equal number of times the subtraction will an- 
nihilate whatever quantity the addition may generate. For 
in fact in the expression -f a — a the sign — represents 
not merely an opposite to + , it indicates at the same 
time the mode in which this opposition is able to operate 
and is to operate, namely by subtraction. If we knew 
nothing of the possibility of such an operation, or if it 
could not be carried out, then w r e could as little evolve 
the result o from a — a, as we could arrive at a result from 
the mere combination in thought of the contradictory 
notions of possibility and impossibility ; in their character 
of opposites these two notions can equally well be repre- 
sented by a and — but this cannot be interpreted by 
a subtraction. 

We see therefore that the proposition a — a = o may 



Chap. V.] 



EXAMPLE OF INTUITION, 



321 



be regarded with equal truth as at once identical and 
synthetic. It is an identical proposition, because it would 
be actually false if the two sides of the equation did not 
represent precisely the same content ; but that the identity 
is there, no mere logical analysis of our — and — signs, 
can possibly inform us ; we learn this solely through the 
immediate intuition of the meaning which the sign — is 
in this particular case capable of bearing, because it is 
related to the increasing or the diminution of quantities. 
Hence the proposition is a synthetic judgment of identity 
between two contents different in form, between a problem 
and its solution. 

A similar instance is presented in the field of mechanics 
by the determination of the resultant of two motions the 
lines of which include an angle. I confine myself here 
to the postulate from which the ordinary attempts at 
demonstration start, namely that where two such forces 
are equal the resultant bisects the angle between them. 
This proposition is commonly regarded as self-evident, 
and we suppose ourselves to possess in this simplest pos- 
sible instance an immediate certainty of a conclusion to 
which any more complicated problems would have to be 
reduced. And undoubtedly the most cautious mind will 
agree to recognise in it not merely a probable hypothesis 
but a truth which only cannot be proved, because it is 
too simple to admit of being proved from anything simpler. 
But the observation which is commonly added by way 
of elucidation, that there is no reason why the resultant 
should approach more nearly to the line of the one force 
than of the other, may serve to illustrate the logical 
character of the proposition in question. For it cannot 
in itself be a positive ground for the necessity of the 
assumed direction of the resultant, that grounds are absent 
for two other classes of directions, unless we start with the 
position that some direction must necessarily be taken, 
and that it cannot coincide with either of the two forces. 

Logic, Vol. II. Y 



322 



THE A PRIORI TRUTHS. 



[Book IT I 



But now it is precisely this that we know from Intuition : 
a merely logical analysis would only teach us that under 
the condition a the element M moves in the direction a, 
under the condition b in the direction ß. Supposing 
both conditions operating together, then M can neither 
move in the direction a nor in the direction ß, because 
either one or the other would suppose one of the two 
conditions entirely inoperative. What then would happen ? 
The two conditions being supposed of equal strength, it 
follows that either both the one and the other must be 
inoperative and M remain at rest, or else both must act 
and be counteracted in equal measure, — supposing always 
that there are ways and means by which that result can 
be brought about. But this last is the important question ; 
that there are such ways and means, and what they con- 
sist in, this is what no method which thought can provide 
is able to inform us. But when we turn on the one hand 
to the Intuition or Perception of space which gives us 
the connexion between the different directions which are 
possible in it, and on the other to the Intuition of motion, 
there it lies all clear before us; there we find that AT 
can satisfy completely both the two conditions at once, 
by so moving as at the expiration of the unit of time / 
to arrive at the same point (being the end of the diagonal 
of the parallelogram) at which it would have arrived in 
two such units of time taken in succession, had it pursued 
first the direction a or ß simply during the one, and then 
the direction ß or a simply during the other; that the 
path finally by which it reaches that point is the diagonal 
itself, follows from the fact that for any small fraction d t of 
the time precisely the same principle holds ; the diagonal 
is the geometric locus of all the points at which M must 
necessarily arrive at the ends of the times d t, 2 d /, $dt 
and the rest. Here again, then, and this time in a pro- 
position of Mechanics, we have a synthetic judgment, 
which establishes the identity between a given problem 



Chap. V.] 



THE 'RATIO LEGIS.' 



and its solution through the instrumentality of immediate 
Intuition. 

362. This for the present must suffice ; I glance at the 
more advanced part of mechanics for a different purpose. 
Whereas its first beginnings by their very simplicity render 
formal methods of demonstration impossible, the problems 
later on become so complicated, that the solutions, although 
strictly following from those fundamental principles, do 
nevertheless, owing to the large variety of the points of 
relation which have to be kept in sight, necessitate very 
lengthy and circuitous processes of abstraction and cal- 
culation. Now indisputable as are the conclusions which 
are thus arrived at, yet nowhere has the desire been more 
keenly felt than in this exact science, to dispense with 
the scaffolding of the Calculus and reduce the results 
obtained to simple conceptions which only need the help 
of computation so far as is involved in their application 
to the conditions, determinate in respect of quantity, which 
particular cases present. I would only remind my readers 
of Gauss' principle of least constraint, which expresses in 
the most universal form the law of all motion as follows : 
a system of material points, however connected with each 
other and whatever may be the external limitations by 
which they are controlled, moves at every instant in the 
greatest possible accordance with the free movement of 
the points, or under the least possible constraint; taking 
as measure of the constraint which the whole system 
endures in every minute portion of time the sum of the 
products obtained by multiplying the square of the de- 
viation of each point from its free movement into the 
mass of that point. 

The second clause in this law supplies the general con- 
ception which is expressed in the first with the mathematical 
form by help of which, for every individual case, the purport 
of what the conception requires is precisely defined and 
made applicable to the given quantitative relations of that 

Y 2 



3 2 4 



THE A PRIORI TRUTHS. 



[Book III. 



case ; but in the first clause we are convinced that we 
possess not merely a general rule which is found as a 
matter of fact to hold good, but the veritable ratio legis, 
from which all the special laws of the various kinds of 
motion are derived. Applying it to the simplest case of all, 
the case of the resultant of two lines of motion, we have 
seen (§232 seq.) that various chains of reasoning will lead 
us with equal certainty to our conclusion. These forms 
of proof however serve only with greater or less cogency to 
co?istrain belief ; on the other hand the reflexion that the 
motion in the line of the diagonal is that by which both 
motive impulses are completely satisfied, and in which no 
part of either is lost, presents itself to us, when once we 
comprehend it and find experience to confirm it, as a 
ground of judgment of an entirely different order, and of 
quite peculiar significance, which arouses in us at once the 
conviction that in it we possess not merely one of the rules 
by the light of which it is admissible to regard the order of 
events observed, but the supreme principle by which they 
are actually governed. 

I added advisedly that we are obliged to presuppose the 
preliminary corroboration of our principle in experience ; 
and in fact however convincing the proposition might be in 
itself, that the conflict of all motions is always so ordered 
that in the final result no element in the effects aimed at by 
the constituent is lost, still without such corroboration it 
would be of very doubtful validity. It would represent a 
principle after which we ourselves perhaps should order the 
world, if the task could be set us, and provided always that 
it was possible and that we had found the means really to 
carry out in every individual case the universal postulate 
which the principle contains. But that the actual world of 
reality or even that the world of thought does possess the 
particular content, form and constitution, and the particular 
combination of elements which renders it possible to unite 
under this single supreme principle all the particular events 



Chap. V.] IDENTITY AS THE GOAL, 



325 



which take place in it, or even the several laws which 
abstract reasoning has presented to us as necessities of 
thought — this we learn only at the end of our journey. 

We know how often in the history of mechanics attempts 
have been made to connect the entire course of the physical 
universe with some such supreme philosophical law; we 
have heard of the constant sum of motion in the universe, 
of the indestructibility of force, of a principle of least action, 
and of a law of Parsimony. All these attempts did not 
merely express the aspiration after a fundamental and self- 
evident idea from which the individual laws, mathematically 
determinable, which govern events, might be derived ; they 
tell us something also of the direction in which the desired 
end is to be looked for. But it has never been found 
possible to determine distinctly and precisely, without 
superfluity or omission, of what subjects of relations so 
universal a conception could be enunciated as no less 
universally valid. 

How far up to the present time any advances have been 
made in this direction, I have not now to enquire \ all that 
I desired to emphasise was the fact of the eager ambition 
displayed by the mind to perfect the circle of its knowledge 
by the aid of principles of the most comprehensive order — 
principles once again which affirm in the form of synthetic 
judgments, which are nevertheless self-evident and univer- 
sally valid, a connexion between two terms of a relation 
whose connexion no process of logical demonstration can 
show to be of an analytical or identical nature. 

363. The final goal of knowledge is usually represented 
in different terms from this. What is aimed at is the 
reduction of all connexions which appear synthetic in 
character to an analytic form — more properly expressed to 
the form of identity — and we are even believed to be 
actually on the way to the consummation of that end. 

At the commencement of our knowledge, we are told, a 
conception £ is made up at first of the small number of 



326 



THE A PRIORI TRUTHS. 



[Book III. 



marks P QR, which we have already found to be connected 
together ; then supposing that fresh experience presents in 
a particular instance a further characteristic Z conjoined 
with S, the proposition S is Z which gives expression to the 
observation made, is considered to be a synthetic judgment. 
If however this new fact of experience becomes established 
as obtaining in all cases of S without exception, Z is 
adopted forthwith into S, and the proposition S is Z has 
now with the enlarged signification of S become analytical. 
This in fact, it is said finally, is the goal towards which all 
knowledge is striving — to reduce those connexions of subject 
and predicate which at first appear so completely synthetic 
to this analytic form, that is to say, to resolve coexistence 
into coherence. And this is a perfectly correct description 
of the origin and growth of knowledge — for it must, alas ! 
be confessed that beyond this point it seldom advances — 
yet it has to be remarked that this ideal described in the 
last words of the sentence is one w T hich is attained only 
to a very modest extent, and that in the sense of the initial 
appearance of mere coexistence giving place to an intelligent 
apprehension of a self-evident law of connexion it is never 
attained at all. 

If we had formed the conception of body to begin with 
out of the qualities of extension, impenetrability, and inertia 
alone, characteristics from which the necessity of mutual 
attraction does not follow, the proposition £ Body is heavy ' 
would undoubtedly have been a synthetic proposition ; but 
the same proposition does not become analytic, even if we 
take into the conception of body the universally observed 
fact of gravitation ; this last property is just as little to be 
derived from the others as it was before, and therefore just 
as much synthetically connected with them as it was in the 
first judgment which expressed that association as a constant 
fact. Undoubtedly we are able, taking this synthetic con- 
junction of all the different marks of as our datum, to 
submit them to the analytic method, and bring them one 



Chap. V.] SYNTHESIS MA Y BE SELF-EVIDENT. 327 

after another before the mind as separate objects of thought; 
but this recognition of the mere fact of constant coexistence 
where the coherence is not understood is in fact the renun- 
ciation of knowledge ; the mind could only rest satisfied if 
the conjunction of any two such properties of S were a sure 
guarantee of the necessary presence of any third. And such 
demonstrations we are able to some extent to establish, and 
whenever we succeed in doing so it means that an advance 
in knowledge has been accomplished ; but it is clear that 
no such result is possible, unless in the last resort we assume 
at some point or other a premiss of the form A + B = C, 
that is to say, a premiss which does not merely affirm 
identity of what is the same 1 by the bare principle . of 
Identity 2 , but affirms identity of the different where no 
reduction to the principle of Identity is possible. Thus the 
supposed transformation of all synthetic knowledge into 
analytical resolves itself after all into the enquiry, what are 
the simplest forms of synthetic truth ? 

364. This contention, though it may perhaps be said to 
amount to nothing more than a needless change of phraseo- 
logy, will nevertheless be in the end admitted. But it will 
be urged in addition that this very necessity of allowing 
certain synthetic combinations to start with, proves the 
inability of human knowledge really to come to a final 
resting-place, and to obtain insight into the inner connexion 
of the coexistent ; everywhere there remains a residue of 
mere facts, of which the connexion of one with another is 
unintelligible, and vouched for only by experience. I cannot 
agree with this opinion, according to which we attain to 
knowledge only where we can affirm exact identity of what 
is exactly identical. For after all whence comes the con- 
fidence with which we hold the proposition A = A to be an 
intelligible truth, except from the immediate self-evidence 
with which it forces itself upon us, and which leaves us no 
room to wish for any mediate demonstration of its certainty 

1 [' Gleiches einander gleich setzt.'] 2 [' Princip der Identität.'] 



3 28 



THE A PRIORI TRUTHS. 



[Book III. 



besides ? But how it happens, by what means it is brought 
about, or from what inner coherence in the nature of things 
I it follows that A is identically like itself, we do not know, 
nor will anyone believe that there is any meaning in asking 
such a question at all. 

If then a perfectly simple synthetic proposition of the 
form A -f B = C presents itself to us with a like degree of 
self-evidence to recommend it, why should a question be 
raised in this instance which was meaningless in the previous 
one ? Why should the latter act of equation only be allowed 
to be valid by the help of some intermediary process, to show 
us how C can =A + H, when in the former our intelligence 
was satisfied to know the fact that A — A? I will not again 
insist on the point that in the processes of our thought no 
such mediation could come from the mere law of Identity, 
that it would always have to begin with a proposition 
A 1 -f B l = C\ analogous in character to the one to be 
proved — for this reflexion would certainly not meet the 
complaints of the incompleteness of a knowledge which is 
said to be incapable of attaining to any supreme self-evident 
principle. But how are we to understand the requirement 
that we should accept some such synthetic connexion as 
given, as valid in itself, and only not accessible to our 
intelligence ? Are we prepared to assume that as a matter 
of fact M, and N are always conjoined in reality without 
affecting one another in any way? But if this is impossible, 
and if it is at the same time impossible that out of one and 
the same self-identical A two different results M and N 
should arise, what else is left to us but to suppose that 
there do exist in the real world certain natural and original 
connexions between things different, original syntheses the 
members of which are not joined together by any inter- 
mediate links, so that the tie between them could appear as 
even the most distant consequence of the law of Identity, 
yet are none the less immediately and really connected? 
If then in the world of Being this must necessarily be the 



Chap, v.] THE SPECULATIVE IDEAL. 329 

case, how can it be demanded of knowledge that it should 
exhibit the certainty and the intelligibility of a given relation 
through a process of mediation which does not exist in that 
relation itself? 

Thus then there may certainly be synthetic truths of an 
ultimate and absolutely simple nature, which as conceived 
in their purest and most simple form possess a validity 
guaranteed not merely by fact but by their own self-evidence, 
a self-evidence however which if we insist on grounding all 
logical truth on the principle of Identity, must no longer be 
called logical but aesthetic, and which accordingly will find 
the touchstone of its validity no longer in the unthinkable- 
ness but in the plain absurdity of its contradictory. To this 
class of truths belong the simplest principles of mechanics ; 
that we regard them together with all truths of like kind 
with them not as the earliest constituents of our knowledge 
which have been there from the beginning, but as its final 
results, to be won only with difficulty and labour, has been 
explained above in terms sufficiently clear to make the 
repetition of it here superfluous. 

365. Special lines of enquiry lead in the first instance to 
single truths of this nature, each one its own evidence and 
standing in no need of support from others. At the same 
time nothing prevents us from bringing them as members 
of one and the same world into connexion with each other 
and searching for a single supreme principle in which they 
may find their unity, just as each one of them had already 
supplied a centre of unity for a body of connected facts. 
It is possible that many such truths may lose in consequence 
their independent value, and that even logical analysis may 
reduce them to particular cases of a more general law, which 
we have found conceptions of a sufficiently comprehensive 
and exalted order of abstraction to express. It is just as 
possible and more likely that the self-evidence with which 
the coherence of the many single elements of truth enables 
them to be ranged under a single fundamental idea, may 



330 



THE A PRIORI TRUTHS. 



rest upon that very same kind of aesthetic propriety on 
the strength of which the single laws themselves were 
formulated, affirming connexions which logic could not 
prove. 

/ Such a development of synthetic truths out of a single 
supreme principle — a development itself synthetic and yet 
at each step necessary — was perhaps the problem of the 
Platonic Dialectic, though as yet but dimly presaged ; it 
may be truly regarded as the end towards which the Hegelian 
revival of the Platonic scheme was directed. From these 
ambitions by which Germany was once inspired, our own 
age has passed with much sobriety to the order of the day, 
to that unremitting labour of empirical enquiry, the incom- 
pleteness of w r hich paralysed the audacious flight of the 
Hegelian idealism. Nor was this the only defect of that 
idealism : unquestionably it was also wrong in regarding 
that which can only be the ultimate goal of a knowledge 
approaching towards completion, as already attained or 
attainable. But in view of the universal idolatry of experi- 
ence which prevails at present, and which is all the cheaper 
and all the safer now that the importance and indispensable- 
ness of its object are visible to all mankind, I will at least 
close with the avowal that I hold that much reviled ideal of 
speculative intuition to be the supreme and not wholly un- 
attainable goal of science, and with the expression of my 
hope that German philosophy will always arouse itself afresh, 
with more of moderation and reserve, yet with no less 
enthusiasm, to the endeavour, not merely to calculate the 
course of the world, but to understand it. 



APPENDIX 1 . 



237.. In the case we have just considered, a very plausible 
supposition, viz., the resolution of a motion, led to a correct 
result, though the conditions of that result really lay in quite 
another field ; there are other cases in which a correct 
though not quite complete supposition leads to results 
which are apparently wrong but which can be made right 
by interpretation. Let a heavy rod whose length is 2 a 
and weight j^lean against a perfectly smooth vertical wall 
and make with the perfectly smooth horizontal plane on 
which it stands the angle (p. It will necessarily slip down 
unless the foot which tends to move away from the wall 
encounters some lateral check. The amount of this resist- 
ance, or, which is the same thing, of the thrust S exerted 
against it by the sliding rod, is expressed by the equation 
W 

S— — cot cp. If the rod stand upright, (p = 90 0 , cot (p = o, 
2 

and therefore S= o ; the rod balances itself freely upon its 
foot, exerts no horizontal pressure at all, and needs no 
lateral check, and can dispense with the vertical wall. As 
(p diminishes, i.e. as the rod slopes, cot (p increases, and 
with it the thrust ; but when (p becomes o, and the rod lies 
horizontally upon the ground, the thrust according to the 
formula is infinitely great, while a glance at the facts show 
us that it must be nothing. This apparent contradiction is 
easily removed. When we propounded the problem we 
thought of course of a continuous horizontal plane capable 
of resistance stretching from the foot of the rod to the 
vertical wall ; but this part of the supposition did not enter 

1 [See Editor's Preface.] 



33 2 



APPENDIX, 



at all into the small calculation by which we arrived at the 

W 

formula S= — cot $ ; here we thought only of the single 

point at the foot which had to carry the weight of the rod ; 
between this point and the wall lay nothing that this calcu- 
lation took count of. In other words the general formula 
treats the two w r alls simply as geometrical loci, of which in 
calculating each particular case we consider only the two 
points at a distance of 2 a from each other upon which the 
forces in question act in this case. Now, if we do not 
go beyond what is involved in this calculation, at the 
moment when $ becomes equal to o there is a gap between 
the foot of the rod and the vertical wall, equal to the length 
of the rod, and through which it would fall when there is no 
perpendicular force to support its weight. It can now no 
longer be said to exert a thrust S; but S signified not only 
this thrust but also the horizontal force, which in the first 
place counteracts this thrust, but which also forms the only 
obstacle that prevents the rod from slipping down into the 
horizontal position in which its weight no longer meets any 
resistance. Now that S becomes infinite when </> becomes 
equal to o means that a force acting horizontally towards the 
vertical wall would have to be infinitely great in order to 
prevent the rod from falling through the gap ; in other 
words, as infinite forces are never found, there is no hori- 
zontal force that could produce that result. We must not 
be misled by the fact that in practice this result is often 
attained by squeezing bodies together in a horizontal direc- 
tion ; for this result is then due to the roughness of the 
surfaces with which the squeezing and the squeezed bodies 
come into contact, and to the compressibility of the latter 
which by slight alterations in its shape furnishes points of 
support which before were wanting. 



INDEX. 



Abstraction, 41, 205. 
Accident and Substance, 78. 
'Actual Reality' {Reale Wirk- 
lichkeit) ; see Reality, ii. 286. 
Adjective, 17-8. 
1 Ad contradictoriam? 109. 

* Ad sub alternant em] 107, ii. 29. 
1 Ad sub alt er nat am] 109. 

' Ad subcontrariam ,' 109. 
Affirmative Judgments, 63. 
Analogy, 137, 322, 328. 

— ' strict,' 328. 

— incomplete, ii. 29. 

— and Induction, cpd., ii. 33. 

— and Hypothesis, ii. 95. 
Analysis, opp. Synthesis, ii. 167. 
Analytical Judgments, 82, 130. 

' Anderssein ' explained, 264. 
Anschauung (' Intuition ' or f Per- 
ception'), note, ii. 309. 
' An-sich-sein ' explained, 264. 
Apagogic (indirect) proof, 307, 

Apodeictic Judgments, 65. 
Apodosis, 123. 

'A posteriori"* Synth. Judgments, 
82. 

* A priori"* Synth. Judgments, 82, 

ii. 295. 

— Ideas, ii. 229. 

— meaning of, ii. 309 ff. 
Apprehension (Synthesis of), 37. 

— Immediate ; see Intuition. 
Archimedes, 319. 
Aristotelian Sorites, 128. 
Aristotle, 53, 54, 64, 114, 12T, 

134, 256, 297. 
■ — (on Platonic Ideas), ii. 215. 
Assertoriai Judgments, 65. 
Augustine, ii. 226. 
Axioms, 301. 



Begriff-, see Concept. 
' Being for itself ' {Für-sich-sein 
265. 

' Being in itself {an-sich-sein), 
264. 

Bernoulli's method, 316. 
Boole, 277 ff. 

Calculus of Chances, ii. 109 ff. 

— Logical, 277 ff. 
Categorical Judgment, 72, ii. 37. 
Categorical character of Classifica- 
tion, 188. 

Cause, opp. Conditions, 125. 

— opp. Ground, ii. 37. 

' Cessante causa — ,' ii. 43. 
Chains of Inference, 128. 
Circle, the proper definition of, 
216. 

Circulus in definiendo, 214. 
Cir cuius in demonstrando, ii. 2. 
Clara perceptio, 222. 
Classification, 163. 

— by combination, 163, cp. 278. 

— natural, 178. 

— by development, 180 ff. 
' Cogito, ergo sum] ii. 226. 
Coherence of world of Thought, 

9 1 » 95- 
Coherent (Ideas), t. 
Coincident (Ideas), I. 
Collective proof, 315. 
Colours (names of), 28. 
Combinatory classification, 164 ff. 
Comparison, 27. 

Compensation in Abstraction, 42. 
Concept, 157 ff. 

— (prior to Judgment), 23, cp. ii. 
27. 

— Imperfect, 39. 

— (' notio') y 44. 



334 



INDEX. 



Concept, Singular, 46. 

— Logical, 47. 

— Universal, 51, ii. 264. 
Conditions, 89. 

— opp. Cause, 125. 
Conjunctions, 21. 
Consequence, 89, 125. 

— opp. Effect, ii. 37. 
Constitutive Concept, 154, 168. 
Construction, 207, 305. 
Content (of a concept, materia) , 

44, 51, and see pp. 15-6. 

— (presentation of to mind), ii. 
259- 

— has ' Validity,' ii. 211, 269. 
Contradiction, 101. 

— principle of, 80. 
Contradictory opposition, 108. 
Contraposition, 112. 
Contraries, 143. 
Contrariety, 101. 

Contrary opposition, ic8. 
1 Conversio pur a] 109 ff. 

— 1 impura] 109. 
Coordinated, 44. 
Coordinated cases, ii. 111. 
Copula, 58, 61. 

— of Categorical Judgment, 78 ff. 
Copulative Judgment, 99. 

— Premisses, 126. 
Current of Ideas, 1 . 

Darwin, 248. 

De die to simpliciter, ii. 5. 

De dicto secundum quid, ii. 5. 

Deductio ad absurdum, 319. 

Deductive proof, 308 ff. 

' Definiendo angustior] 215. 

— ' latior, 1 215. 
Definitio, 211. 
Definition, 209. 

— Nominal and Real, 213 ff. 

— Genetic, 222. 
Descartes, ii. 179, 226. 
Description, 208. 
Descriptive Definition, 221. 
Sfurepa ovaia, ii. 216. 
Dialectic, Plato's, ii. 222, 330. 

— Hegel's, 191, ii. 330. 
Siacpopa, 53. 

Dictum de omni, 100, 330. 



Differentiae specific ae, 57, 209. 
Ding {see Thing and Sache), has 
existence, ii. 208. 

— opp. ' Content ' and * objective ' 
{sachlich), but = 'external 
object' {Gegenstand), ii. 270. 

Direct Proof, 307. 
Disjunction, Incomplete, ii. 12. 

— and Probability, ii. no. 
Disjunctive Judgment, 68, 99, 102. 

— Law of Thought, 102, 330. 

— Premisses, 126. 
Disparates, 104, 143, 149, 228. 
Distinct a perceptio, 222. 
Distinction, 26. 

Duality, the Principle of, 286. 

Effect, opp. Consequence, 94, ii. 

38. 

' Effectuiun naturalium] ii. 47-8. 
€i8os, 51. 

Eleatic dialectic, ii. 201. 
Ellipse, equation of, 172. 
Episyllogism, 128. 
Equal ; see ' Like.' 
Equation of the Ellipse, 172. 
Equations, constitutive, 154. 
Equilibrium, problem of conditions 

of, 336. 
Essential marks, 161. 
Evidence, transmission of, ii. 117. 
Excluded Middle, principle of, 

100, 102. 
Existence, Judgment of, 74. 

— opp. Validity, ii. 209, 267. 
Expectation, mathematical, ii. 123. 

— moral, 376, ii. 125. 
Experiment, ii. 39. 
Explanation, opp. Classification, 

188. 

— Incomplete, ii. 10. 
Explanatory theory, 188. 
Extent (of a concept, e ambitus'^. 

44, 5 1 - 

Fact ; see Sache. 

— and Law, ii. 79. 

— knowledge of {Erkenntniss der 
Sache}, ii. 186. 

Fallacia falsi medii, ii. 4. 
False premisses, ii. 1. 



INDEX. 



335 



Fichte's Zeitschrift quoted, 100. 
Fiction, dist. Hypothesis, ii. 90. 
Figures of Syllogism, 116, 123. 
Figure, second, 134. 

— third, 137. 

' Für-sich-sein' explained, 265. 

Galen, figure of, 120. 

Gauss (principle of least con- 
straint), ii. 323. 

' General ' (Generic) opp. Uni- 
versal Judgment, 97. 

General term (mathematical), ii. 

79- 
yivos, 51. 
Genus, 44, 51. 
Genus fir oximum, 209. 
i Geschichte der Aesthetik in 

Deutschland' (Author's), 262. 
Goklenius, Goklenian Sorites, 128. 
Grammar cpd. with Logic, 20. 
Gravity, 90. 

Ground and Consequence, ii. 37. 
Ground, ii. 38. 

Hegel, 47. 54» 2Ö2 - 

Hegelian Dialectic, 262, ii. 350. 

Heraclitus, ii. 200. 

Herbart, 94. 

Hume, ii. 295. 

VTTOTvrrüjaus, Pyrrhonian, ii. 196. 
Hypotheses, subsidiary, ii. 100. 
Hypothesis, dist. Postulate, ii. 90. 
Hypothetical Judgment, 65, 89, 
99, ii. 37, 68. 

— Syllogism, 123. 

— character of Explanation, 188. 
Hysteron Proteron, ii. 3. 

Idea (' Idee' see ii. 202 note), 168, 
ii. 202 ff. 

— (' Vorstellung'), meaning of, 31. 
Ideal, 180. 

Ideas, Coherent and Coincident, 1. 

— opp. Impressions, 13. 

— (opp. Begriff), ii. 265. 
Identical {'gleich'), ii. 254. 
Identity {'Identität '), principle of, 

99, ii. 296, 329. 

— (' Gleichheit'), ii. 260, and see 
note, ii. 254. 



idiov, 54. 

Immediate Inference, 106 ff. 

Imperfect concepts, 39. 

Impersonal Judgment, 72. 

Impressions (opp. ' Ideas'), 13. 

Incommensurables, Inference con- 
cerning, 148. 

Indirect (apagogic) proof, 307, 
317'. 

Induction, cpd. w. Analogy, 136 
ff., ii. 33. 

— Complete, 315. 

— Imperfect, ii. 32. 
' Inductive,' 308. 

' Inductive Logic/ ii. 22. 
Inference and Disjunction, 106. 

— Immediate, 106. 

Innate Ideas {'Ideen'), ii. 229 

ff., 311 ff. 
Intuition {Anschauung), ii. 309 

ff. 

Inverse Ratio of Extent and Con- 
tent, 51. 

1 Investigation of the Laws of 

Thought' (Boole's), 277 ff. 
Ionians, philosophy of, 252. 

Jevons, 297. 

Judgment (and Concept), 23, 56. 

— (Real significance of), ii. 273. 

' Justification ' of a conception, 
3°4- 

Kant, Classification of Judgments, 
61 ff. 

— on Categorical Judgment, 78 

— Analytical and Synthetical 
Judgments, 82, 130. 

— Synthetical Judgments, 84. 

— Synthetical Judgments, a 
firiori, ii. 295. 

— his Categories, ii. 220. 

— Matter and Form, ii. 233. 
KCLTrjyopeiv, 77« 

Kepler, ii. 69. 

Language, a logical, 272. 

Laplace, ii. 131. 

Law, meanings of, ii. 68. 

— and Rule, ii. 71. 

— of Nature, ii. 68. 



33^ 



INDEX. 



Law, and Facts, ii. 83. 

— Faith in, ii. 292. 
Leibnitz, 272. 

Like ('gleich'); see note, p. 327, 

and note, ii. 254. 
Limitative Judgments, 6?>. 
Line without mass, 352 ff. 
Linnaean classification, 166. 
Locke on Innate Ideas, ii. 311 ff. 
Logical ; a logical language, 272. 

— the calculus, 277 ff. 

Major premiss, 130. 
Mark, meaning of the term, 47. 
Marks, essential and unessential, 
161. 

Material ('sachlich,' opp. 'for- 
mal"), 279. 

Mathematical Truth and Law of 
Identity, ii. 297. 

Mathematics and Logic, 35, 147. 

Mean, arithmetical, ii. 143. 

Measurements, correction of, ii. 
H3. . 

Mechanical Physics, 90. 
Metaphysics (Aristotle's), ii. 215. 
Method of least squares, ii. 145. 
Middle concept, 114. 
Middle term, ambiguity of, ii. 4. 
Minor premiss, 133. 
Minority, the ' winning minority,' 
ii. 151. 

Modality of Judgments, 64, 112. 

— True, 66. 

' Modus ponendo ponens,' etc., 125. 
Moods, 117 ff. 

Naming, 25. 

Natural Classification, 178. 
Negative Judgments, 63. 
Newton, ii. 47-8. 

V01)TOS TOTTOS, Ü. 2 1 3. 

Nominal Definition, 2 1 3. 
Nominalism, ii. 267. 
Numbers and Things (Pythagorean 
doctrines), 255-6. 

Objectification, 14, 140. 

Objective (' objectiv '), Objectivity 

(' objectivität'), 14-5, cp. ii. 

-ori 
y 



Objective (' objectiv J opp. ' subjec- 
tiv'), ii. 279. 

— (sachlich), ii. 253, 260. 
Observation, ii. 38. 
Occurrence, opp. Existence, ii. 

208, 289. 
c Operationskreis des Logikcalcüls ' 

(Schröder), 278 ff. 
Opposition of Judgments, 108. 
' Order of the day,' ii. 161. 

OplOt/JLOS, 211. 

' Other being ' (' Anders-sein 1 , 

264. 
ovaia, ii. 211. 

Parallelogram of forces, proofs of, 

355 ff. 
Paralogisms, ii. 14. 
Part of a concept, 47. 
Particular Judgments, 62, 84. 
Pendulum's swing, time of, ii. SS. 
Perception (synthesis of), 37. 

— and the Impersonal Judgment, 

Personal error, ii. 74. 
Petitio principii, ii. 2. 
Philosophical calculus (Leibnitz , 
272. 

Picture in the mind, opp. Con- 
ception, ii. 26, cp. ii. 264. 
Plato, 56, 76, ii. 210. 

— (the Meno), ii. 312. 

— (the Theaetetus), ii. 201. 
Platonic ' Ideas,' ii. 202 ff. 

— * Dialectic,' ii. 330. 
Polylemmas, 127. 
Porphyrius, 54. 
Position, 26. 

Postulate, dist. Hypothesis, ii. 90. 
Predicate, 58, 77. 
Premisses, it 4. 

— Hypothetical, 126. 

— Establishment of, 133. 
Prepositions, 21. 
Presentation, the act of, ii. 259. 
Principium exclusi tertii, 100, 102. 
Principles of Analogy discussed, 

322 ff. 

'Principles of Science' (Jevons 1 . 
297. 

Probability, ii. 10;. 



INDEX. 



337 



Problematic Judgments, 65, 68 ff. 
Progressive proof, 307. 
Projections, ii. 76. 
Proof, forms of, 307 ff. 
Proofs by exclusion and by limita- 
tion, 318. 
Property, 54, 78. 

— and Thing, 78. 
Proportion, Inference by, 150 ff. 
Prosyllogism, 128. 
Protagoras, ii. 201. 

Protasis, 123. 
TTpojTTj ovaia, ii. 216. 
Proving too much, ii. 10. 
Proving too little, ii. 10. 
* PseudomenosJ ii. 17. 
Psychology, ii. 332. 
' Pure case/ ii. 37, 67. 
Pythagoras, 251 ff. 

Quality of Judgments, 63. 
Quantification of the Predicate, 
297. 

Quantitative variations of Causes 

and of Effects, ii. 61 ff. 
Quantitative Ideas, 33. 
Quantitative Judgments, 83. 
Quantity of Judgments, 62. 
' Quaternio terminorumj 115, 

ii. 4. 

Real, Reality; see Actual {Wirk- 
lich, Wirklichkeit), ii. 208, cp. 
PP- 14-5. 

— (of external world, ' real'), ii. 

187, 286. 

— ('real,' opp. 'formal'), ii. 279. 
Real Definition, 213. 

Realism, ii. 267. 

'Reason' and 'Condition;' see 

' Ground,' 93. 
Receptivity of Thought, 35. 

— and Spontaneity, 36, ii. 233. 
Reduction, 122. 

— to impossibility, 122. 
Refutation, 314. 
Regressive proof, 307. 
Rehnisch, 100. 
Relations, reality of, ii. 260. 
Remotive Judgment, 99. 

— Premisses, 126. 
Logic, Vol. II. 



Resolution of forces into com- 
ponents, 364. 
Rotatory forces, 340, 347. 
Rule, ii. 71. 

Sache, see Ding, ' of those things 
{Sachen), the given thinkable 
contents,' ii. 273. 

Sachlich, see ' Material,' ' Objec- 
tive,' ' matter of knowledge 
{sachlich), although not things 
(dinghaß),' ii. 268, see note, 
ii. 270. 

Schröder, ' Ope7'ationskreis des Lo- 

gikcalciils,' 278 ff. 
Self- evidence, may be spurious, 

301. 

Sensations, disparate groups of, 
228 ff. 

Sextus Empiricus, ii. 183. 
Similar, Similarity (' Aehnlich,' 

' Aehnlichkeit'), defined, 327. 
Singular Concepts, 45. 
Singular Judgments, 62, 70. 
Socrates (^in Plato), 224, ii. 202. 
Sophisms, ii. 14. 
Sophists, ii. 201. 
Sorites, 128. 

Space, of more than three dimen- 
sions, 231. 
Spatial Intuition, ii. 310. 
Species, 44, 50. 

— Perfect and Imperfect, 170. 

— of two genera, 1 70. 
Specific difference, 53. 
Spectrum analysis, ii. 51. 
Speculative Thought, 195. 
State, 54. 

Statistics, ii. 84, 139. 
Subalternation, 107. 
Subcontrary opposition, 108. 
Subject, 58, 76. 

— Grammatical and Logical, 60. 
Subordination, 44, 49. 
Substance and Accident, 78. 
Substantive, 17. 
Substitution, Inference by, 14,-;. 
Subsumed, 44. 
Subsumption, 49. 

— Inference by, 129. 
Sufficient Reason, Law of, 91. 



338 ' INDEX. 



av^ßcßrjKos, 54. 
Syllogism, 114. 

— Subsumptive, 129. 

— Aristotelian, criticised, 135. 

— Real significance of, ii. 276. 
Symbols, 146. 

Synthesis of Apprehension, 37. 

— of Perception, 37. 

— opp. Analysis, ii. 167. 
Synthetical Judgments, 82, 84, 

131. 

— (a priori), ii. 295. 



Tabula rasa, ii. 232. 
Taylorian Theorem, 369-374. 
Things, the peculiar 'reality' of, 

14 ff., cp. ii. 208. 
Thing and Property, 78. 



Thought (opp. c current of ideas'), 
3, 13. 

Tpo-noi, sceptical, ii. 194. 
Type, 179. 

Ultimate Concepts, 55. 
Universal (' first Universal '), 30. 

— Judgments, 62, 97. 

— Laws, 186. 

— conception, ii. 265. 
Universals which are not concepts, 

46, 53- 

Universality and Necessity, ii. 240. 

Validity, ii. 209, 267. 
Verb, 17-8. 

' Vorstellung;* see Idea. 
Zeno, on motion, ii. 14. 



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phytology from the Standpoint of the 
Botanist. By H. Graf zu Solms- 
Laubach. Translated and revised 
by the same. Royal 8vo, half- 
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OXFOKD HISTOEY OF MUSIC, 

8vo. Edited by W. H. Hadow, M.A. 
The Polyphonic Period. Part I 

(Method of Musical Art, 330-1330). 

By H. E. Wooldridge, M.A. 15s. net. 
The Seventeenth Century. By Sir 

C. H. H. Parry, M.A. , D. Mus. 15s.net. 
The Age of Bach and Handel. By 

J. A. Fuller Maitland, M. A. 15s. net. 

IN PREPARATION. 
The Polyphonic Period. Part II. 

By H. E. Wooldridge, M.A. 
The Viennese School. By W. H. 

Hadow, M.A. 
The Romantic Period. By E. 

Dannreuther, M.A. 



OXFORD 
AT THE CLARENDON PRESS 
LONDON, EDINBURGH, AND NEW YORK 
HENRY FROWDE 



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